Factoring Polynomials

Overview

Factoring polynomials is a crucial skill in algebra. Factoring involves rewriting a polynomial as a product of simpler polynomials, which makes solving equations and simplifying expressions easier. There are different methods for factoring polynomials depending on the number of terms and the type of polynomial.

Common Methods of Factoring

  • Factoring out the greatest common factor (GCF): If all terms in the polynomial share a common factor, factor that out first.
  • Factoring by grouping: Group terms in pairs and factor each pair separately.
  • Factoring trinomials: Factor quadratics of the form \( ax^2 + bx + c \).
  • Difference of squares: Factor expressions like \( a^2 - b^2 \) into \( (a + b)(a - b) \).
  • Perfect square trinomials: Factor expressions like \( a^2 + 2ab + b^2 \) into \( (a + b)^2 \).

Factoring by Grouping Example

Consider the polynomial \( x^2 + 5x + 2x + 10 \). First, group the terms:

\( (x^2 + 5x) + (2x + 10) \)

Now, factor each group:

\( x(x + 5) + 2(x + 5) \)

Finally, factor out the common binomial \( (x + 5) \):

\( (x + 5)(x + 2) \)

The factored form of \( x^2 + 5x + 2x + 10 \) is \( (x + 5)(x + 2) \).

Factoring Trinomials Example

Consider the trinomial \( x^2 + 7x + 10 \). We need to find two numbers that multiply to 10 and add to 7. These numbers are 2 and 5. So, the factored form is:

\( (x + 2)(x + 5) \)

Difference of Squares Example

The difference of squares can be factored as \( a^2 - b^2 = (a + b)(a - b) \). For example:

\( x^2 - 9 = (x + 3)(x - 3) \)

Perfect Square Trinomial Example

A perfect square trinomial takes the form \( a^2 + 2ab + b^2 = (a + b)^2 \). For example:

\( x^2 + 6x + 9 = (x + 3)^2 \)

Practice Questions

  1. Question 1: Factor the polynomial \( x^2 + 6x + 8 \).
    Solution

    We need to find two numbers that multiply to 8 and add to 6. These numbers are 2 and 4. The factored form is:

    \( (x + 2)(x + 4) \)

  2. Question 2: Factor the polynomial \( x^2 - 16 \).
    Solution

    This is a difference of squares. We can factor it as:

    \( (x + 4)(x - 4) \)

  3. Question 3: Factor the polynomial \( x^2 + 9x + 20 \).
    Solution

    We need to find two numbers that multiply to 20 and add to 9. These numbers are 4 and 5. The factored form is:

    \( (x + 4)(x + 5) \)

  4. Question 4: Factor the polynomial \( 2x^2 + 8x \).
    Solution

    The greatest common factor is 2x. We factor out 2x:

    \( 2x(x + 4) \)

  5. Question 5: Factor the polynomial \( x^2 + 10x + 25 \).
    Solution

    This is a perfect square trinomial. We can factor it as:

    \( (x + 5)^2 \)

→Solving Polynomial Equations