7.2 Operations with Polynomials: Add, Subtract, Multiply, Divide & Factor

Why This Section Matters

Polynomials extend everything from Section 7.1 to expressions with more than one term, where variables can now carry exponents greater than 1. This section covers indicators A.1.d through A.1.g — still part of Assessment Target A.1, but the piece of it that deals with polynomials rather than purely linear expressions. It is also the direct on-ramp to quadratic equations, tested later in Assessment Target A.4 (Chapter 9): you cannot solve x² + 5x + 6 = 0 by factoring unless you can already factor x² + 5x + 6 as a polynomial. Binomial multiplication and trinomial factoring taught here are load-bearing prerequisites for at least two later assessment targets (A.4 quadratic equations and A.7 quadratic function evaluation), so gaps in this section compound in later chapters.

Core Vocabulary

• A polynomial is a sum of terms, each built from a number times a variable raised to a whole-number, non-negative exponent (no negative or fractional exponents).
• A monomial has one term (5x², −3). A binomial has two terms (x + 4). A trinomial has three terms (x² + 5x + 6).
• The degree of a term is the exponent on its variable (x² has degree 2; x has degree 1; a constant like 7 has degree 0). The degree of the polynomial is the highest degree among its terms.

Adding and Subtracting Polynomials (A.1.d, part 1)

Add or subtract polynomials by combining like terms — terms with the same variable raised to the same power — across both polynomials.

(3x² + 2x − 5) + (x² − 4x + 7) = 4x² − 2x + 2

For subtraction, distribute the negative sign across every term of the polynomial being subtracted before combining:

(3x² + 2x − 5) − (x² − 4x + 7) = 3x² + 2x − 5 − x² + 4x − 7 = 2x² + 6x − 12

The single most common error on this skill: flipping the sign of only the first term of the second polynomial (getting 3x² + 2x − 5 − x² − 4x + 7, which is wrong) instead of flipping the sign of all three terms.

Multiplying Polynomials — FOIL (A.1.d, part 2)

To multiply two binomials, use FOIL: First, Outer, Inner, Last — a mnemonic for which pairs of terms to multiply together.

(x + 4)(x − 6):

• First: x × x = x²
• Outer: x × (−6) = −6x
• Inner: 4 × x = 4x
• Last: 4 × (−6) = −24

Combine the Outer and Inner terms (−6x + 4x = −2x) for the final result: x² − 2x − 24.

To multiply a monomial by a polynomial with any number of terms, distribute the monomial across every term:

3x(2x² − 5x + 1) = 6x³ − 15x² + 3x

Dividing Factorable Polynomials (A.1.d, part 3)

GED items test two division methods:

1. Divide each term by a monomial divisor. (10x² − 15x) ÷ 5x: divide 10x² by 5x to get 2x, and −15x by 5x to get −3, giving 2x − 3.
2. Factor and cancel for binomial divisors. (x² − 9) ÷ (x − 3): factor the numerator as a difference of squares, (x + 3)(x − 3), then cancel the shared factor (x − 3) with the divisor, leaving x + 3 (valid for x ≠ 3).

Evaluating Polynomial Expressions (A.1.e)

Substitute the given integer for every occurrence of the variable, in parentheses, then follow order of operations. The classic trap here involves exponents on negative numbers: (−2)² = 4, because the entire quantity −2 is squared, but −2² (without parentheses grouping the negative sign) would mean −(2²) = −4.

Evaluate 3x² − 2x + 1 at x = −2:

3(−2)² − 2(−2) + 1 = 3(4) − (−4) + 1 = 12 + 4 + 1 = 17

A common wrong answer, 5, results from computing 3(−2)² as (3 × −2)² = 36 or from squaring before applying the coefficient incorrectly — always square the substituted value first, then multiply by the coefficient.

Factoring Polynomials (A.1.f)

Three factoring patterns cover nearly every GED item on this indicator:

For trinomial factoring, always check the sign pattern: if c is negative, p and q have opposite signs; if c is positive and b is negative, both p and q are negative.

Writing Polynomial Expressions from Context (A.1.g)

Polynomial expressions often model area. A rectangular garden with length (x + 5) and width x has area x(x + 5) = x² + 5x. A composite scenario — a square garden of side x surrounded by a walkway that adds 2 feet to each dimension — models the outer area as (x + 4)(x + 4) = x² + 8x + 16, since 2 feet is added on both sides of each dimension.

Exam Scenario

A GED item gives the dimensions of a rectangular deck as (x + 3) feet by (x − 2) feet and asks for a polynomial expression representing its area. Multiplying using FOIL: (x + 3)(x − 2) = x² − 2x + 3x − 6 = x² + x − 6. Expect distractor choices such as x² − x − 6 (sign error on the Outer/Inner combination) and x² + x + 6 (sign error on the Last term).