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Step 11: Algebra 1

Polynomials and factoring

Step 11 · Algebra 1 topics

Add, subtract and multiply polynomials, take out common factors, and factor trinomials and differences of squares.

Adding and multiplying polynomials

A polynomial is a sum of terms such as 3x², −5x and 7. To add or subtract polynomials, combine like terms: terms with the same power of x.

Example: Adding

Simplify (3x² − 5x + 2) + (x² + 4x − 7).

  1. x² terms: 3x² + x² = 4x².
  2. x terms: −5x + 4x = −x.
  3. Numbers: 2 − 7 = −5.

Answer4x² − x − 5

Example: Subtracting

Simplify (5x² + 2x) − (2x² − 3x + 1).

  1. The minus changes every sign in the second bracket: 5x² + 2x − 2x² + 3x − 1.

Answer3x² + 5x − 1

To multiply, multiply every term in one polynomial by every term in the other. An area model keeps track of the pieces.

(x + 2)(x + 3): the four boxes add to x² + 5x + 6.

Example: Multiplying binomials

Expand (x + 2)(x + 3).

  1. x · x + x · 3 + 2 · x + 2 · 3.
  2. x² + 3x + 2x + 6.

Answerx² + 5x + 6

Common factors

Factoring undoes multiplying. Always start by taking out the greatest common factor (GCF) of all the terms.

Example: A common factor

Factor 6x³ + 9x².

  1. The GCF of 6 and 9 is 3, and of x³ and x² is x².
  2. Take out 3x².

Answer3x²(2x + 3)

Example: Grouping

Factor x³ + 2x² + 5x + 10.

  1. Group in pairs: x²(x + 2) + 5(x + 2).
  2. Now x + 2 is a common factor.

Answer(x + 2)(x² + 5)

Example: A difference of squares

Factor x² − 49.

  1. x² − 49 is x² − 7².
  2. a² − b² = (a + b)(a − b).

Answer(x + 7)(x − 7)

Factoring trinomials

To factor x² + bx + c, find two numbers that multiply to c and add to b.

Example: Both positive

Factor x² + 7x + 12.

  1. Pairs that multiply to 12: 1 and 12, 2 and 6, 3 and 4.
  2. 3 + 4 = 7.

Answer(x + 3)(x + 4)

Example: A negative constant

Factor x² − 2x − 15.

  1. The numbers multiply to −15 and add to −2: −5 and 3.

Answer(x − 5)(x + 3)

Example: A number in front

Factor 2x² + 7x + 3.

  1. Multiply 2 × 3 = 6. Find numbers that multiply to 6 and add to 7: 6 and 1.
  2. Split the middle: 2x² + 6x + x + 3.
  3. Group: 2x(x + 3) + 1(x + 3).

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

Try it yourself

Work each one on paper first, then open the answer to check.

  1. Simplify (2x² + 3x − 1) − (x² − x + 4).

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    x² + 4x − 5

  2. Expand (x − 3)(x + 5).

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    x² + 2x − 15

  3. Expand (2x + 1)².

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    4x² + 4x + 1

  4. Factor 10x² − 15x.

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    5x(2x − 3)

  5. Factor x² − 81.

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    (x + 9)(x − 9)

  6. Factor x² + 9x + 20.

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    (x + 4)(x + 5)

  7. Factor 3x² − 10x − 8.

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    (3x + 2)(x − 4)

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