Polynomials and Factoring

Grade 9 · Math · Study Guide

Master polynomial operations and factoring techniques including GCF, trinomials, and special products.

A polynomial is an algebraic expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, with non-negative integer exponents. Examples include monomials (one term), binomials (two terms), and trinomials (three terms).

Factoring a polynomial means rewriting it as a product of simpler polynomials. The first step is always to factor out the Greatest Common Factor (GCF). For example, 6x² + 9x = 3x(2x + 3).

For trinomials of the form ax² + bx + c where a = 1, look for two numbers that multiply to c and add to b. For example, x² + 5x + 6 = (x + 2)(x + 3) because 2 × 3 = 6 and 2 + 3 = 5.

Special patterns speed up factoring. The difference of squares: a² − b² = (a + b)(a − b). Perfect square trinomials: a² + 2ab + b² = (a + b)². Recognizing these patterns saves time and reduces errors.

When a ≠ 1, use the AC method: multiply a and c, find factors of that product that add to b, split the middle term, then factor by grouping. Always verify by expanding your answer.

Key terms

Polynomial
An expression with one or more terms involving variables raised to non-negative integer powers
GCF (Greatest Common Factor)
The largest factor that divides evenly into all terms of a polynomial
Difference of Squares
The pattern a² − b² = (a + b)(a − b) used to factor certain binomials
Perfect Square Trinomial
A trinomial of the form a² ± 2ab + b² that factors as (a ± b)²
Factor by Grouping
A method of factoring polynomials with four terms by grouping pairs and factoring each pair
Leading Coefficient
The coefficient of the term with the highest degree in a polynomial

Flashcards

Factor 6x³ − 12x²
6x²(x − 2)
Factor x² − 25
(x + 5)(x − 5) — difference of squares
Factor x² + 6x + 9
(x + 3)² — perfect square trinomial
Factor x² − 8x + 15
(x − 3)(x − 5)
Factor 3x² + 7x + 2
(3x + 1)(x + 2)
Factor x² − x − 12
(x − 4)(x + 3)
Degree of 4x³ − 2x + 7
3 (highest exponent)
Factor 4x² − 16
4(x² − 4) = 4(x + 2)(x − 2)

Questions

  1. Which of the following is the completely factored form of 2x² + 8x?

    1. 2(x² + 4x)
    2. 2x(x + 4)
    3. x(2x + 8)
    4. 2x(x) + 8x
    Show answer

    Answer: 2x(x + 4)

    Factor out 2x from both terms: 2x² + 8x = 2x(x + 4). Check: 2x · x = 2x², 2x · 4 = 8x.

  2. Factor x² − 9 completely.

    1. (x − 3)²
    2. (x + 3)(x − 3)
    3. (x − 9)(x + 1)
    4. (x + 3)²
    Show answer

    Answer: (x + 3)(x − 3)

    x² − 9 is a difference of squares (a² − b² where a = x, b = 3), so it factors as (x + 3)(x − 3).

  3. The expression x² + 4 can be factored using the difference of squares pattern.

    Show answer

    Answer: False

    Difference of squares requires subtraction: a² − b². Since x² + 4 uses addition, it cannot be factored over the real numbers using this pattern.

  4. Factor x² + 7x + 12. The factored form is (x + 3)(x + __).

    Show answer

    Answer: 4

    Find two numbers that multiply to 12 and add to 7: 3 × 4 = 12 and 3 + 4 = 7. So x² + 7x + 12 = (x + 3)(x + 4).

  5. Factor 2x² + 5x + 3 completely.

    1. (2x + 1)(x + 3)
    2. (2x + 3)(x + 1)
    3. (x + 3)(2x − 1)
    4. (2x − 3)(x − 1)
    Show answer

    Answer: (2x + 3)(x + 1)

    Using the AC method: 2 × 3 = 6. Factors of 6 that add to 5 are 2 and 3. Rewrite: 2x² + 2x + 3x + 3 = 2x(x + 1) + 3(x + 1) = (2x + 3)(x + 1).

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