Divide any polynomial p(x) by d(x) using the long division algorithm — shows each subtraction step with remainder.
p(x) = d(x) · q(x) + r(x)
Step 1: Divide the leading term of p(x) by the leading term of d(x) to get the first term of q(x).
Step 2: Multiply that term by the entire divisor d(x).
Step 3: Subtract from the current dividend. Bring down the next term.
Repeat until the degree of the remainder is strictly less than the degree of d(x).
r = p(c) when d(x) = x − c
When you divide p(x) by (x − c), the remainder equals p(c). This is the Remainder Theorem.
If the remainder is 0, then d(x) divides p(x) evenly — d(x) is a factor of p(x). This is the Factor Theorem.
Use synthetic division as a faster shortcut when the divisor is linear (degree 1).
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