Polynomial Operations

ACT Math · Axiom Academy

Add, subtract, multiply, divide, and factor polynomials — the four operations the ACT tests over and over, all built from one idea: combining like terms. 1. Polynomials — and Combining Like Terms A polynomial is a sum of terms, each a coefficient times a power of x . The degree is the highest exponent; the leading coefficient is the number in front of that highest-degree term. For 3x^2 + 2x - 5 : degree is 2, leading coefficient is 3. Adding: combine matching powers of x Subtracting: distribute the minus sign first 2. Multiplying — FOIL as an Area To multiply two binomials, use FOIL : First, Outer, Inner, Last. Picture the product as the area of a rectangle split into four smaller rectangles — each partial product is one piece of that area, and FOIL just names the four pieces in order. Multiply the first term of each binomial. Multiply the outermost pair of terms. Multiply the innermost pair of terms. Multiply the last term of each binomial. Two special patterns worth memorizing Square of a binomial: (a+b)^2 = a^2 + 2ab + b^2 — the middle term is always twice the product of the two pieces. Difference of squares: (a+b)(a-b) = a^2 - b^2 — the middle terms cancel completely, so the result has only two terms. 3. Dividing Polynomials — Synthetic Division When you divide a polynomial by (x - r) , synthetic division turns long division into three repeated steps: bring down, multiply by r , add. Divide x^3 - 2x^2 - 5x + 6 by (x - 3) — here r = 3 .

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