Polynomials and Factoring Study Guide: GED Math (page 3)

FOIL

To multiply binomials, you must multiply each term by every other term and add the products. The acronym FOIL can help you remember how to multiply binomials. FOIL stands for first, outside, inside, and last.

Example

(3x + 1)(7x + 10) =

3x and 7x are the first pair of terms, 3x and 10 are the outermost pair of terms, 1 and 7x are the innermost pair of terms, and 1 and 10 are the last pair of terms.

Therefore, (3x)(7x) + (3x)(10) + (1)(7x) + (1)(10) = 21x² + 30x + 7x + 10.

After you combine like terms, you are left with the answer: 21x² + 37x + 10.

Factoring

Multiplying the binomials (x + 1) and (x + 2) creates the quadratic expression x² + 3x + 2. That expression can be broken back down into (x + 1)(x + 2) by factoring.

A quadratic trinomial (a trinomial is an expression with three terms) that begins with the term x² can be factored into (x + a)(x + b). Factoring is the reverse of FOIL. Find two numbers, a and b, that multiply to the third value of the trinomial (the constant) and that add to the coefficient of the second value of the trinomial (the x term).

Given the quadratic x² + 6x + 8, you can find its factors by finding two numbers whose product is 8 and whose sum is 6. The numbers 1 and 8 and the numbers 4 and 2 multiply to 8, but only 4 and 2 add to 6. The factors of x² + 6x + 8 are (x + 2) and (x + 4). You can check your factoring by using FOIL: (x + 2)(x + 4) = x² + 4x + 2x + 8 = x² + 6x + 8.

What are the factors of 2x² + 9x + 9?

This quadratic will be factored into (2x + a)(x + b). Find two numbers that multiply to 9. Two times one of those numbers plus the other must equal 9, the coefficient of the second term of the quadratic trinomial. The numbers 1 and 9 and the numbers 3 and 3 multiply to 9. Two times 3 plus 3 is equal to 9, so the factors of 2x² + 9x + 9 are (2x + 3) and (x + 3).

Removing a Common Factor

If a polynomial contains terms that have common factors, the polynomial can be factored by using the reverse of the distributive law.

Look at the binomial 49x³ + 21x. The greatest common factor of both terms is 7x.

Therefore, you can divide 49x³ + 21x by 7x to get the other factor.

Factoring 49x³ + 21x results in 7x(7x² + 3).

Practice problems for these concepts can be found at:

Algebra and Functions Practice Problems: GED Math