Algebra 1 · Chapter 6 of 8

Polynomials and Factoring

Add, multiply, and factor polynomial expressions, with special attention to common factors and quadratic patterns.

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Why this chapter matters

Factoring reveals zeros and structure that are hidden in expanded form, making later equation solving and graph analysis possible.

What you will learn

  • Classify polynomials by degree and number of terms.
  • Add and multiply polynomials accurately.
  • Factor greatest common factors, trinomials, and differences of squares.

Understand the core ideas

Polynomials are sums of terms with nonnegative integer exponents, and their structure matters. Degree tells the highest exponent and predicts long-run graph behavior in later courses. When adding or subtracting polynomials, combine only matching powers. When multiplying, distribute each term in one polynomial across each term in the other, then collect like powers. Organizing terms in descending degree reduces mistakes and makes patterns easier to spot.

Factoring reverses multiplication and exposes hidden structure. First pull out the greatest common factor, because this reduces coefficient size and can reveal a known pattern. For trinomials x2+x^2 + bx + c, search for two numbers that multiply to c and add to b. For a2−b2a^2 - b^2, use (a - b)(a + b). Verify by multiplying your factors back to the original polynomial. That quick check catches sign errors before they become equation-solving errors, and it also confirms whether your factorization is complete or still missing a shared factor.

Key terms

degree
The highest exponent of the variable in a polynomial.
monomial
A single-term polynomial, such as 6x26x^2.
greatest common factor
The largest factor shared by all terms in an expression.
difference of squares
A pattern a2−b2a^2 - b^2 that factors as (a - b)(a + b).

Factor a quadratic completely

Factor 6x2+15x−366x^2 + 15x - 36.

  1. Factor out the greatest common factor 3: 6x2+15x−36=3(2x2+5x−12)6x^2 + 15x - 36 = 3(2x^2 + 5x - 12).
  2. For 2x2+5x−122x^2 + 5x - 12, multiply 2 and -12 to get -24.
  3. Find two numbers that multiply to -24 and add to 5: 8 and -3.
  4. Rewrite and group: 2x2+8x−3x−12=2x(x+4)−3(x+4)2x^2 + 8x - 3x - 12 = 2x(x + 4) - 3(x + 4).
  5. Factor the common binomial: (x + 4)(2x - 3), then include the outside 3.
Result: The complete factorization is 3(x + 4)(2x - 3).

A common misconception

Claim: Factoring is optional because expanded form is always easier to use.

Correction: Expanded form is useful for arithmetic, but factoring reveals zeros and intercepts and is essential for solving many equations.

Lessons in this chapter

  1. Polynomial operationsCombine and multiply polynomial terms without mixing degrees.
  2. Greatest common factorsRemove the largest shared factor before using other patterns.
  3. Factoring trinomialsReverse multiplication to express a quadratic as two binomials. Read the full guide →
  4. Special productsRecognize perfect squares and differences of squares.

Study task

Build and simplify an expression for the area of a rectangular garden with sides x + 3 and x + 5, then factor the expanded result back.

Chapter checkpoint

Factor x2+7x+12x^2 + 7x + 12.

Find two numbers that multiply to 12 and add to 7: 3 and 4. The factorization is (x + 3)(x + 4).

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