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Grades 9-12

We learned how to factor by grouping in lesson 9.04 Factoring by grouping . We will factor trinomial expressions in this lesson using the same factoring application.

**Trinomials** can be rewritten as polynomials with four terms and factored by grouping.

Consider the polynomial expressions factored by grouping below:

- What patterns do you notice between the original expression and the terms used to rewrite the linear term?
- Choose one of the linear terms and rewrite the term in a different way than shown, then determine whether the polynomial can still be factored by grouping.

When using the grouping method to factor a trinomial, the coefficients of the terms used to rewrite the linear term have a sum equivalent to the linear coefficient from the original polynomial and a product equivalent to the product of the trinomial's leading coefficient and constant.

Steps in factoring a quadratic trinomial of the form ax^{2} + bx + c:

Factor out any GCF.

(If a is negative, we can also divide out a factor of -1 before continuing.)

Find two numbers, r and s, that multiply to ac and add to b.

Rewrite the trinomial with four terms in the form ax^{2} + rx + sx + c.

Factor by grouping.

Check whether the answer will not factor further and verify the factored form by multiplication.

Remember to include any common factors divided out at the start, so each step results in an equivalent expression.

Factor x^{2} + 10 x - 24.

Worked Solution

Factor 3 x^{2} - 27.

Worked Solution

Factor 5 x^{2} - 18x + 9.

Worked Solution

Idea summary

Steps in factoring a quadratic trinomial:

Factor out any GCF.

(If a is negative, we can also divide out a factor of -1 before continuing.)

Find two numbers, r and s, that multiply to ac and add to b.

Rewrite the trinomial with four terms, in the form ax^{2} + rx + sx + c.

Factor by grouping.

Check whether the answer will not factor further and verify the factored form by multiplication.

Remember to include any common factors divided out at the start, so each step results in an equivalent expression.