Polynomial Functions, Division, and Zeros

Discussion prompt

Before we open Polynomial Functions, Division, and Zeros: without looking back, what was the main idea of Combining Functions, Composition, and Inverses, and what could you do by the end of it that you could not do before?

Hint: One sentence for the idea, one for the skill. If the second one is blank, that is the part to revisit.

Discussion prompt

A polynomial function is a sum of terms, and every term is a number times a whole-number power of the variable.

That is stated as though it always holds. Do one of two things: produce a case where it fails, or say precisely what rules such a case out. "It just does" is not on the menu.

Hint: Hunt at the extremes first — zero, one, negative, empty, equal. If every extreme survives, the reason they survive is the proof.

Sort into buckets

Sort these by Polynomial?, from memory, without looking back at What does not count. Telling them apart on the spot is the skill; the table is only where the answer happens to be written down.

yes

f(x) = 4x^3 - 2x + 7; f(x) = 5

no

f(x) = 3/x + 1; f(x) = sqrt(x) + 2; f(x) = 2x^(-1) + x

g1

Polynomial? is "yes" for f(x) = 4x^3 - 2x + 7, f(x) = 5 — that is what the table on "What does not count" records, and it is the single property separating this group from the rest.

g2

Polynomial? is "no" for f(x) = 3/x + 1, f(x) = sqrt(x) + 2, f(x) = 2x^(-1) + x — that is what the table on "What does not count" records, and it is the single property separating this group from the rest.

Sort into buckets

Every line below is part of the definition of leading term or of leading coefficient — one or the other, never both. Put each where it belongs.

leading term

The term carrying that highest power.; Here the leading term is 4 times x to the fifth.

leading coefficient

The number multiplying the leading term.; Its SIGN is what decides half of the graph's story.

b1

The term carrying that highest power. Here the leading term is 4 times x to the fifth.

b2

The number multiplying the leading term. Here it is 4. Its SIGN is what decides half of the graph's story.

Animation

Shows: Degree and leading sign decide the ends — a rendered Manim animation.

Rendered with Manim.

Takeaway: Odd degree means the ends go opposite ways; the sign says which.

Discussion prompt

Explain Polynomial graphs are smooth and unbroken by analogy to something with no College Algebra in it at all — a queue, a recipe, a map, a bank balance, whatever fits. Then say where your analogy breaks.

Hint: An analogy that never breaks is not an analogy, it is the same idea wearing a hat. Find the seam — that is the part that is actually new.

Step through it

Step through For huge inputs, the leading term swallows everything one row at a time. What is driving the change, and what would the row after the last one be?

1. Step 1: x is 10
2. Step 2: x is 100
3. Step 3: x is 1,000

Step through it

Step through The four end-behavior cases one row at a time. What is driving the change, and what would the row after the last one be?

1. Step 1: Degree is even
2. Step 2: Degree is even
3. Step 3: Degree is odd
4. Step 4: Degree is odd

Discussion prompt

Explain Writing end behavior in arrow notation to a student a year behind you. No notation, no jargon they have not met — and it still has to be true.

Hint: If your explanation needs a symbol they have never seen, you are describing the notation rather than the idea.

Put in order

Put the moves of Worked example: end behavior of a cubic into the order they have to happen.

1. Find the degree and the leading coefficient
2. Degree three is odd, so the arms disagree
3. The leading coefficient is negative, so flip the plain cubic
4. Verify with one big positive and one big negative input

Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. It is already in standard form, so the first term written is the leading term.

Animation

Shows: Each line of the worked example "end behavior of a cubic", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: At ten the output is negative one thousand four hundred sixty-one, so the right arm is heading down. At negative ten the output is positive two thousand four hundred fifty-nine, so the left arm is heading up. Both match the prediction.

Predict first

A student writes this, and it looks reasonable:

The polynomial arrives out of order and the student grabs whatever is written first.

It is wrong. Say what breaks — and say it before you turn the page.

Correct: It is the first thing on the page, so it feels like the leading term.

Rewrite in standard form first. Highest power leads, always.

Why: It is the first thing on the page, so it feels like the leading term.

Comparison matrix

From Trap: the first term written is not always the leading term: refill the f(x) column from what you know. The rest of the table is as it appeared.

Discussion prompt

Worked example: end behavior from a factored form — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Add the exponents to get the degree

Animation

Shows: Each line of the worked example "end behavior from a factored form", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: At ten the value is negative one hundred seventy-seven thousand nine hundred fifty-seven, so the right arm goes down. At negative ten the value is positive forty-one thousand five hundred three, so the left arm goes up. The prediction holds.

Put in order

These are the steps of Pattern: reading end behavior in three moves, scrambled. Put them back in order before the next slide shows you.

1. Get the leading term. Rewrite in standard form, or multiply the leading pieces of the factors together.
2. Even or odd degree? Even means the arms agree; odd means they disagree.
3. Positive or negative leading coefficient? Positive leaves the right arm up; negative flips the whole picture.

Why: This is the order the recipe itself gives. Recalling the sequence without the slide in front of you is the difference between recognising the method and being able to run it — most of what goes wrong in practice is a step done out of turn.

Eliminate the wrong options

What is the end behavior of this function?

3 of these 4 are wrong. Strike them one at a time, and say what rules each one out before you strike the next. The survivor is the answer.

• A. Up on the left, down on the right
• B. Up on both ends
• C. Down on the left, up on the right
• D. Down on both ends

Survives elimination: A

Why: In standard form this is negative three times x to the seventh, plus five x squared, plus two. The leading term has degree seven (odd, so the arms disagree) and a negative coefficient (so the right arm goes down). That gives up on the left and down on the right.

Check your understanding

What is the end behavior of this function?

• A. Up on the left, down on the right (correct)
• B. Up on both ends
• C. Down on the left, up on the right
• D. Down on both ends

Answer: A

Why: In standard form this is negative three times x to the seventh, plus five x squared, plus two. The leading term has degree seven (odd, so the arms disagree) and a negative coefficient (so the right arm goes down). That gives up on the left and down on the right.

Why B tempts people

Used five x squared - the first term written - as the leading term instead of the degree-seven term. Always rewrite in standard form first.

Why C tempts people

Got the odd degree right but ignored the negative sign on the coefficient. That negative reflects the whole graph, swapping which arm goes up.

Why D tempts people

Treated degree seven as even, which would make the two arms agree. Seven is odd, so the arms must point in opposite directions.

Comparison matrix

From Zero, x-intercept, root, factor: one idea: every row here is a choice with a cost. Fill the What it means here column, then say which row you would actually pick and what you give up for it.

Predict first

The table runs: 0 | 0 | 0 | 0 | 0 · 6 | 216 | -144 | -72 | 0

In Worked example: find the zeros by factoring, given the rows so far: what is the next one — the row where x is -2?

Correct: -2 | -8 | -16 | 24 | 0

Why: The relationship between the columns, not the individual numbers, is what generates the next row. Every term has an x in it. Pulling it out is always the first factoring move, and it hands you one zero immediately.

Animation

Shows: Each line of the worked example "find the zeros by factoring", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: At zero the output is zero. At six: two hundred sixteen minus one hundred forty-four minus seventy-two is zero. At negative two: negative eight minus sixteen plus twenty-four is zero. All three check out.

Match the pairs

Match each term to the definition this lesson gave it — not the one you would guess from the word.

• t1. degree
• t2. leading term
• t3. leading coefficient
• t4. multiplicity

• d1. The highest exponent on the variable. Here the degree is 5.
• d2. The term carrying that highest power. Here the leading term is 4 times x to the fifth.
• d3. The number multiplying the leading term. Here it is 4. Its SIGN is what decides half of the graph's story.
• d4. The exponent on a factor. Here negative two has multiplicity 2, one has multiplicity 3, and four has multiplicity 1.

Why: These are the working definitions of degree, leading term, leading coefficient, multiplicity as Polynomial Functions, Division, and Zeros uses them. Pairing them correctly is the test of whether you could state each one with the slide switched off.

Animation

Shows: Multiplicity decides touch or cross — a rendered Manim animation.

Rendered with Manim.

Takeaway: The double root touches and turns; the single root passes through.

Discussion prompt

Read the picture before the words. What is this showing, and what is the one thing it is built to make obvious? Commit to an answer, then read on.

Hint: Name the parts, then say what changes between them — and if nothing changes, say what is being held still.

Predict first

A student writes this, and it looks reasonable:

The exponent 2 gets read as a count of crossings.

It is wrong. Say what breaks — and say it before you turn the page.

Correct: The reasoning sounds fine: multiplicity two, so two crossings.

Even multiplicity means the graph touches the axis and turns back.

Why: The reasoning sounds fine: multiplicity two, so two crossings. But a graph cannot cross the axis twice at a single point.

Comparison matrix

From Trap: cross versus touch, backwards: refill the f(x) column from what you know. The rest of the table is as it appeared.

Step through it

Step through How many zeros, how many turns one row at a time. What is driving the change, and what would the row after the last one be?

1. Step 1: Degree is 2
2. Step 2: Degree is 3
3. Step 3: Degree is 4
4. Step 4: Degree is 5

Animation

Shows: How many roots can there be — a rendered Manim animation.

Rendered with Manim.

Takeaway: The Fundamental Theorem of Algebra, stated for this course.

Discussion prompt

Read the picture before the words. What is this showing, and what is the one thing it is built to make obvious? Commit to an answer, then read on.

Hint: Name the parts, then say what changes between them — and if nothing changes, say what is being held still.

Step through it

Step through Worked example: sketch from factored form one row at a time. What is driving the change, and what would the row after the last one be?

1. Step 1: x is -2
2. Step 2: x is 0
3. Step 3: x is 3

Predict first

Same routine, but now there is a stretch factor with a minus sign out front.

Commit before you compute: what does Worked example: a sketch with a negative leading coefficient come out to? A rough magnitude and the right form is enough — the point is to have something concrete to be wrong about.

Correct: Verify by testing one input in each region

Why: A prediction you can defend turns the computation into a check rather than a leap of faith — and an answer that contradicts it is caught on the spot. At negative two the value is ten, above the axis, matching up on the left.

Animation

Shows: Sketching a polynomial — a rendered Manim animation.

Rendered with Manim.

Takeaway: Four steps, and the shape has almost no freedom left.

Predict first

What does the graph do at the zero where x equals 1?

Answer it in your own words, now, with nothing to choose from. The options are on the next slide — and picking the right one off a list is an easier skill than producing it.

Correct: Touches the axis and turns around

Why: The factor that vanishes at one is squared, so its multiplicity is two. An even multiplicity means the factor never changes sign as the input passes through, so the whole product cannot change sign and the graph must turn around.

Check your understanding

What does the graph do at the zero where x equals 1?

• A. Touches the axis and turns around (correct)
• B. Crosses straight through the axis
• C. Crosses the axis, flattening out first
• D. Has a break in the graph there

Answer: A

Why: The factor that vanishes at one is squared, so its multiplicity is two. An even multiplicity means the factor never changes sign as the input passes through, so the whole product cannot change sign and the graph must turn around.

Why B tempts people

The cross-versus-touch rule applied backwards. Odd multiplicity crosses; even multiplicity bounces, and two is even.

Why C tempts people

That is the behavior at negative four, where the multiplicity is three. Odd and at least three means cross with a flattening, but at one the multiplicity is two.

Why D tempts people

Polynomial graphs are smooth and unbroken everywhere. Breaks and vertical asymptotes belong to rational functions, not polynomials.

Check your understanding

At most how many turning points can the graph of a degree-5 polynomial have?

• A. 4 (correct)
• B. 5
• C. 6
• D. 3

Answer: A

Why: A polynomial of degree n has at most n minus one turning points, so a degree-five polynomial turns at most four times. It can have up to five zeros, which is the count students usually reach for by mistake.

Why B tempts people

That is the maximum number of zeros, not turning points. The turning-point cap is one less than the degree.

Why C tempts people

Added one to the degree instead of subtracting one. The formula is degree minus one.

Why D tempts people

Subtracted two from the degree. Between five crossings there are four places the curve must turn, so the cap is four.

Animation

Shows: How many turns a polynomial can make — a rendered Manim animation.

Rendered with Manim.

Takeaway: Degree four turns at most three times — and here it turns exactly three.

Discussion prompt

Worked example: long division with a missing degree — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Insert a placeholder for the missing squared term

Animation

Shows: Each line of the worked example "long division with a missing degree", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: Expanding gives the cubed term, then three x squared minus three x squared cancels, then two x minus nine x is negative seven x, and negative six is the constant. That is the original dividend exactly, so the division is correct.

Fill in the blanks

From Trap: skipping the placeholder for a missing degree — finish the line. Write what belongs on the right of the equals sign before you look.

(x-3)\left(x^2-4x-12\right) - 42 = x^3 - 7x^2 - 6 \;\ne\; x^3 - 7x - 6

Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. With no placeholder there is nothing in the squared column, so the next term slides left to fill it.

Annotate

From Trap: skipping the placeholder for a missing degree — read this one piece at a time. What is each part doing?

On: \( x - 3 \;\overline{)\; x^3 - 7x - 6} \)

• With no placeholder there is nothing in the squared column, so the next term slides left to fill it. From that moment on, every subtraction is happening in the wrong column.
• The arithmetic inside each pass is fine, but the columns were shifted, so the answer belongs to a different problem.
• Multiplying the divisor by that quotient and adding the remainder gives a polynomial with a squared term in it. The original had none, so this cannot be the right quotient.

Predict first

The divisor is not linear here, so long division is the only tool that works.

Commit before you compute: what does Worked example: dividing by a quadratic come out to? A rough magnitude and the right form is enough — the point is to have something concrete to be wrong about.

Correct: Verify by expanding the right-hand side

Why: A prediction you can defend turns the computation into a check rather than a leap of faith — and an answer that contradicts it is caught on the spot. The product gives the fourth-power term, then six x squared minus eleven x squared is negative five x squared, and negative twenty-two plus fourteen is negative eight.

Animation

Shows: Each line of the worked example "dividing by a quadratic", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The product gives the fourth-power term, then six x squared minus eleven x squared is negative five x squared, and negative twenty-two plus fourteen is negative eight. Adding the two x gives back the original dividend exactly.

Animation

Shows: Synthetic division, step by step — a rendered Manim animation.

Rendered with Manim.

Takeaway: The last number is the remainder — zero means you found a factor.

Predict first

Worked example: synthetic division is about to be worked. State your hypothesis first: which rule or definition decides this one, and what is the first move it forces? Then watch whether the example agrees with you.

Correct: Find the number for the box

Why: Set the divisor equal to zero. The value that makes it zero is two, so two goes in the box.

A hypothesis you wrote down is falsifiable; a vague sense of how it will go is not. If the example opens somewhere else, that gap is the thing worth chasing.

Animation

Shows: Each line of the worked example "synthetic division", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: Expanding the divisor times the quotient gives two x cubed minus five x squared plus three x minus two, and subtracting five leaves minus seven as the constant. That is the original dividend exactly.

Predict first

A student writes this, and it looks reasonable:

The divisor has a plus sign, so the student writes the positive number in the box.

It is wrong. Say what breaks — and say it before you turn the page.

Correct: It is the number visible in the divisor, so it feels right.

Solve the divisor for zero and use that value.

Why: It is the number visible in the divisor, so it feels right. But synthetic division adds instead of subtracting, so the box needs the value that makes the divisor zero.

Draw it

Translate both ways. First write the expression above as a sentence with no symbols in it at all. Then cover it, and write your sentence back as notation. If the two versions disagree, the disagreement is the thing to fix.

Predict first

A student writes this, and it looks reasonable:

The divisor has a squared term, but the shortcut is tempting anyway.

It is wrong. Say what breaks — and say it before you turn the page.

Correct: The reasoning copies the linear rule: set the divisor to zero and grab a number.

A divisor of degree two or higher means long division, no exceptions.

Why: The reasoning copies the linear rule: set the divisor to zero and grab a number. But a quadratic divisor has no single such number, and the method quietly divides by something else.

Annotate

From Trap: synthetic division with a divisor that is not linear — read this one piece at a time. What is each part doing?

On: \( q(x) = 3x^2 - 11, \qquad r(x) = 2x + 14 \)

• The reasoning copies the linear rule: set the divisor to zero and grab a number. But a quadratic divisor has no single such number, and the method quietly divides by something else.
• A degree-four polynomial divided by a degree-two one must give a degree-two quotient. Getting a cubic is the tell that the machine divided by the linear factor instead.
• Divide leading term by leading term, multiply, subtract, bring down - exactly as in the earlier example.

Discussion prompt

Run Pattern: which division method to use with this step confiscated:

Divisor linear? Use synthetic, with the value that makes the divisor zero in the box.

Is it still possible? If it is, say what takes its place and what it costs you. If it is not, say exactly what that step was providing that nothing else does.

Hint: A step you can drop for free was never load-bearing. If you cannot drop it, name the thing that goes wrong the moment it is gone.

Comparison matrix

From Pattern: which division method to use: every row here is a choice with a cost. Fill the Number in the box column, then say which row you would actually pick and what you give up for it.

Predict first

What is the remainder, and what does it tell you?

Commit to an answer, then rate it — certain, fairly sure, or guessing — and write the rating down before you turn the page.

Correct: Remainder 0, so x - 3 is a factor

Why: With 3 in the box the bottom row is 1, -1, -2, 0. The last entry is the remainder, so it is zero, and a zero remainder means the divisor divides evenly. Substituting three into the polynomial confirms it: twenty-seven minus thirty-six plus three plus six is zero.

The rating matters as much as the answer: confident-and-wrong is the combination that survives revision, because nothing about it feels like it needs revisiting.

Check your understanding

What is the remainder, and what does it tell you?

• A. Remainder 0, so x - 3 is a factor (correct)
• B. Remainder 0, so x + 3 is a factor
• C. Remainder -60, so x - 3 is not a factor
• D. Remainder 6, so x - 3 is not a factor

Answer: A

Why: With 3 in the box the bottom row is 1, -1, -2, 0. The last entry is the remainder, so it is zero, and a zero remainder means the divisor divides evenly. Substituting three into the polynomial confirms it: twenty-seven minus thirty-six plus three plus six is zero.

Why B tempts people

Confused the number in the box with the factor. The box holds the value that makes the divisor zero, so a box of 3 corresponds to the factor x minus 3, not x plus 3.

Why C tempts people

Put negative three in the box instead of three. That computes the value of the polynomial at negative three, which is negative sixty, and answers a different question.

Why D tempts people

Reported the constant term of the dividend as the remainder. The remainder is the last number in the bottom row of the synthetic table, not the last coefficient you started with.

Animation

Shows: The Remainder Theorem — a rendered Manim animation.

Rendered with Manim.

Takeaway: Which makes synthetic division a fast way to evaluate.

Put in order

Put the moves of Worked example: evaluating with synthetic division into the order they have to happen.

1. List the coefficients, inserting a placeholder for the missing squared term
2. Put negative two in the box and run multiply-then-add
3. Read the last entry as the value
4. Verify by direct substitution

Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. A fourth-degree polynomial needs five slots.

Animation

Shows: The Factor Theorem — a rendered Manim animation.

Rendered with Manim.

Takeaway: Roots and factors are two descriptions of one fact.

Predict first

Show that one is a zero, then find every other zero.

Commit before you compute: what does Worked example: using the Factor Theorem to break a cubic… come out to? A rough magnitude and the right form is enough — the point is to have something concrete to be wrong about.

Correct: Verify by multiplying the factorization back out

Why: A prediction you can defend turns the computation into a check rather than a leap of faith — and an answer that contradicts it is caught on the spot. Expanding gives the cubed term, then five x squared minus one x squared is four x squared, then negative two x minus five x is negative seven x, and the constant is two.

Animation

Shows: Each line of the worked example "using the Factor Theorem to break a cubic down", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: Expanding gives the cubed term, then five x squared minus one x squared is four x squared, then negative two x minus five x is negative seven x, and the constant is two. That is the original polynomial exactly.

Discussion prompt

Explain The Rational Zero Theorem to a student a year behind you. No notation, no jargon they have not met — and it still has to be true.

Hint: If your explanation needs a symbol they have never seen, you are describing the notation rather than the idea.

Discussion prompt

Worked example: listing the candidates — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Collect the divisors of the constant term

Animation

Shows: Each line of the worked example "listing the candidates", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: Every numerator on the list divides six and every denominator divides two, so the list is right. Testing three: fifty-four minus twenty-seven minus thirty-three plus six is zero, so at least one candidate really is a zero.

Predict first

A student writes this, and it looks reasonable:

The list gets copied down and handed in as the set of zeros.

It is wrong. Say what breaks — and say it before you turn the page.

Correct: A degree-three polynomial has at most three zeros.

The list is a set of candidates to test. Test them until one gives zero, then divide and move on.

Why: A degree-three polynomial has at most three zeros. Twelve is impossible, and that contradiction alone should stop the pen.

Step through it

Step through Trap: treating the candidate list as the answer one row at a time. One of these columns never changes — find it, and say why it cannot.

1. Step 1: Candidate is 1
2. Step 2: Candidate is -1
3. Step 3: Candidate is 2

Put in order

These are the steps of Pattern: finding every zero of a cubic or quartic, scrambled. Put them back in order before the next slide shows you.

1. Factor out anything obvious first - a greatest common factor, or a grouping if it is sitting right there.
2. List the rational-zero candidates: divisors of the constant over divisors of the leading coefficient, both signs.
3. Test candidates with synthetic division until a remainder of zero appears.

Why: This is the order the recipe itself gives. Recalling the sequence without the slide in front of you is the difference between recognising the method and being able to run it — most of what goes wrong in practice is a step done out of turn.

Discussion prompt

Pattern: finding every zero of a cubic or quartic works on the cases you have just seen. Push it to the edge: what is the most degenerate input it still handles — empty, zero, one item, everything equal — and what is the first case where it stops being true? Name the case, not just "it breaks".

Hint: Try the smallest legal input, then the largest, then the one where two things collide. Methods are specified at their edges; the middle takes care of itself.

Fill in the blanks

From Worked example: all the zeros of a cubic — finish the line. Write what belongs on the right of the equals sign before you look.

f(x) = 2x^3 - 3x^2 - 11x + 6

Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. From the candidate list, three is a convenient whole number to try.

Animation

Shows: Each line of the worked example "all the zeros of a cubic", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The two linear pieces give two x squared plus three x minus two, and multiplying by the remaining factor gives back the original cubic exactly, so no zero was lost or invented.

Predict first

Which number is NOT on the list of possible rational zeros?

Answer it in your own words, now, with nothing to choose from. The options are on the next slide — and picking the right one off a list is an easier skill than producing it.

Correct: 3/2

Why: The numerator must divide the constant 4 and the denominator must divide the leading coefficient 3. In three halves the numerator 3 does not divide 4 and the denominator 2 does not divide 3, so it fails on both counts and cannot be on the list.

Check your understanding

Which number is NOT on the list of possible rational zeros?

• A. 4/3
• B. 3/2 (correct)
• C. -1/3
• D. 2

Answer: B

Why: The numerator must divide the constant 4 and the denominator must divide the leading coefficient 3. In three halves the numerator 3 does not divide 4 and the denominator 2 does not divide 3, so it fails on both counts and cannot be on the list.

Why A tempts people

Four thirds is on the list: 4 divides the constant 4 and 3 divides the leading coefficient 3.

Why C tempts people

Negative one third is on the list: 1 divides 4, 3 divides 3, and both signs are always included.

Why D tempts people

Two is on the list: 2 divides the constant 4 and the denominator 1 divides the leading coefficient 3.

Check your understanding

What are all the zeros?

• A. 1, 3, -2 (correct)
• B. -1, -3, 2
• C. 1, -3, 2
• D. 1, 2, 3

Answer: A

Why: Testing one gives one minus two minus five plus six, which is zero. Dividing by the matching factor leaves the quadratic x squared minus x minus six, which factors into pieces giving three and negative two. So the zeros are 1, 3, and negative 2.

Why B tempts people

Read the zeros straight off the factors without flipping the sign. A factor of x plus two means the zero is negative two, not positive two.

Why C tempts people

Factored the leftover quadratic as x plus three times x minus two. That expands to x squared plus x minus six, which has the wrong middle sign.

Why D tempts people

Factored the leftover quadratic as x minus two times x minus three. That expands to x squared minus five x plus six, not x squared minus x minus six.

Discussion prompt

Explain The Fundamental Theorem of Algebra by analogy to something with no College Algebra in it at all — a queue, a recipe, a map, a bank balance, whatever fits. Then say where your analogy breaks.

Hint: An analogy that never breaks is not an analogy, it is the same idea wearing a hat. Find the seam — that is the part that is actually new.

Animation

Shows: Where to look for rational roots — a rendered Manim animation.

Rendered with Manim.

Takeaway: A finite list to test, rather than an infinite search.

Comparison matrix

From Exactly n zeros, counting multiplicity: refill the Zeros with multiplicity column from what you know. The rest of the table is as it appeared.

Discussion prompt

One more fact, and it only applies when the coefficients of the polynomial are real numbers.

That is stated as though it always holds. Do one of two things: produce a case where it fails, or say precisely what rules such a case out. "It just does" is not on the menu.

Hint: Hunt at the extremes first — zero, one, negative, empty, equal. If every extreme survives, the reason they survive is the proof.

Fill in the blanks

From Worked example: all the zeros of a quartic — finish the line. Write what belongs on the right of the equals sign before you look.

f(x) = x^4 - x^3 + 7x^2 - 9x - 18

Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. The leading coefficient is one, so the candidates are just the divisors of eighteen with both signs.

Animation

Shows: Building a polynomial from its zeros — a rendered Manim animation.

Rendered with Manim.

Takeaway: One more point pins down the leading coefficient.

Discussion prompt

Worked example: build a polynomial from a complex zero — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Bring the conjugate along

Animation

Shows: Each line of the worked example "build a polynomial from a complex zero", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: At four: sixty-four minus ninety-six plus fifty-two minus twenty equals zero. And the complex zero satisfies the quadratic factor, since one minus two i squared is negative three minus four i, minus two times one minus two i is negative two plus four i, and adding five gives zero.

Put in order

Put the moves of Worked example: build a polynomial through a given point into the order they have to happen.

1. Write the factored form with an unknown leading coefficient
2. Substitute the given point
3. Solve for the leading coefficient
4. Expand into standard form
5. Verify the point and one of the zeros

Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. Three zeros means three linear factors and degree three.

Animation

Shows: Sign charts from the factors — a rendered Manim animation.

Rendered with Manim.

Takeaway: The sign can only change at a root of odd multiplicity.

Eliminate the wrong options

Which degree-3 polynomial with real coefficients and leading coefficient 1 has zeros 4 and 1 minus 2 times the imaginary unit?

3 of these 4 are wrong. Strike them one at a time, and say what rules each one out before you strike the next. The survivor is the answer.

• A. x^3 - 6x^2 + 13x - 20
• B. x^3 - 6x^2 + 5x + 12
• C. x^2 - 5x + 4
• D. x^3 + 6x^2 + 13x + 20

Survives elimination: A

Why: Real coefficients force the conjugate to be a zero too. Multiplying the conjugate pair gives x squared minus 2x plus 5, and multiplying that by x minus 4 gives x cubed minus 6x squared plus 13x minus 20. Substituting 4 gives 64 minus 96 plus 52 minus 20, which is zero.

Check your understanding

Which degree-3 polynomial with real coefficients and leading coefficient 1 has zeros 4 and 1 minus 2 times the imaginary unit?

• A. x^3 - 6x^2 + 13x - 20 (correct)
• B. x^3 - 6x^2 + 5x + 12
• C. x^2 - 5x + 4
• D. x^3 + 6x^2 + 13x + 20

Answer: A

Why: Real coefficients force the conjugate to be a zero too. Multiplying the conjugate pair gives x squared minus 2x plus 5, and multiplying that by x minus 4 gives x cubed minus 6x squared plus 13x minus 20. Substituting 4 gives 64 minus 96 plus 52 minus 20, which is zero.

Why B tempts people

Multiplied the conjugate pair as 1 minus 4 instead of 1 plus 4, forgetting that the square of the imaginary unit is negative one. That gives x squared minus 2x minus 3 instead of x squared minus 2x plus 5.

Why C tempts people

Dropped the imaginary part entirely and used 1 as a real zero, which leaves only two factors and a degree-2 answer. The problem asked for degree 3.

Why D tempts people

Wrote each factor as x plus the zero instead of x minus the zero. Flipping every sign flips the zeros to negative 4 and negative 1 plus or minus 2 times the imaginary unit.

Draw it

One page, no notation unless you need it: draw how these connect — What a Polynomial Is · End Behavior: the Two Arms · Zeros and Multiplicity · Dividing Polynomials · Synthetic Division · Remainder, Factor, and Rational Zeros. Put an arrow wherever one of them is what makes another possible, and label the arrow with why.