The Factor Theorem

Pre-Calculus · Axiom Academy

Zeros and factors are the same fact wearing two hats — find one, and you've found the other. Take P(x) = x^3 - 6x^2 + 11x - 6 . To hunt for factors, we test values of c : plug each into P and watch where it lands. A marker walks the x -axis and drops a probe to the curve — wherever the curve touches the axis , P(c)=0 , and a factor (x-c) snaps into place. 2. Why P(c)=0 Makes It a Factor Saying " c is a zero" and " (x-c) is a factor" are linked by division . Divide P(x) = x^3 - 2x^2 - 5x + 6 by (x-3) using synthetic division. Watch each coefficient drop, get multiplied by c=3 , and add into the next column. The final cell is the remainder — and it lands on 0 . Each running value is multiplied by c=3 and added to the next coefficient. The last cell is what's left over. Zero means the division is exact. No remainder with nothing left behind. The top row of results is q(x) = x^2 + x - 2 , the cofactor. 3. The Biconditional on the Graph The Factor Theorem runs both ways . Plot P(x) = (x-1)(x-2)(x-3) . Every place the curve crosses the x -axis is a zero — and each crossing peels off a factor (x-c) . Three crossings, three factors: the complete factorization assembles itself as the curve draws. 4. Reverse It: Build a Polynomial From Roots If a zero gives a factor, then choosing the zeros builds the polynomial. Want roots at ? The Factor Theorem says use the factors (x-1) , (x+2) , (x-4) — multiply them and you've engineered a polynomial that hits zero at exactly those spots.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.