Evaluate P(c) by direct substitution or synthetic division (Horner's method) — and check if c is a root.
P(x) = ((aₙx + aₙ₋₁)x + ...)x + a₀
Direct substitution evaluates each term independently: compute xⁿ, multiply by its coefficient, sum all results. For degree n this uses up to n(n+1)/2 multiplications.
Horner's method (synthetic substitution) restructures the polynomial as nested multiplications so that only n multiplications and n additions are needed — the minimum possible for degree n.
This is faster and accumulates less floating-point rounding error, which matters whenever you evaluate polynomials repeatedly (e.g. inside Newton's method or numerical integration).
The synthetic tableau is a visual record of Horner's algorithm: bring down the leading coefficient, multiply by c, add the next coefficient — repeat until done. The final number is P(c).
P(c) = remainder when P(x) ÷ (x − c)
When you divide P(x) by the linear factor (x − c), the remainder is exactly P(c). This is the Remainder Theorem.
Synthetic division simultaneously divides P(x) by (x − c) and evaluates P(c) — the last bottom-row entry is P(c) (the remainder), and the other bottom-row entries are the coefficients of the quotient polynomial.
Factor / Root corollary: P(c) = 0 if and only if (x − c) divides P(x) evenly with no remainder — meaning c is a root and (x − c) is a factor.
One-on-one tutoring builds the intuition for evaluation, factoring, and roots — we work through your actual homework so the connections between direct substitution, synthetic division, and the Remainder Theorem click into place.