Education.com
Try
Brainzy
Try
Plus

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

By
Updated on Mar 23, 2011

Practice problems for these concepts can be found at:

Algebra and Functions Practice Problems: GED Math

### Polynomials

A polynomial is the sum or difference of two or more unlike terms. For example, look at 2x + 3yz. This expression represents the sum of three unlike terms: 2x, 3y, and –z. There are three kinds of polynomials:

1. A monomial is a polynomial with one term, as in 2b3.
2. A binomial is a polynomial with two unlike terms, as in 5x + 3y or 5x3y.
3. A trinomial is a polynomial with three unlike terms, as in y2 + 2z – 6.

### Operations with Polynomials

To add polynomials, be sure to change all subtraction to addition and change the sign of the number that was being subtracted to its opposite. Then, combine like terms.

Example

(3y3 – 5y + 10) + (y3 + 10y – 9)

Change all subtraction to addition and change the sign of the number being subtracted: 3y3 + –5y + 10 + y3 + 10y + –9.

Combine like terms: 3y3 + y3 + –5y + 10y + 10 + –9 = 4y3 + 5y + 1.

If an entire polynomial is being subtracted, change all of the subtraction to addition within the parentheses and then add the opposite of each term in the polynomial.

Example

(8x – 7y + 9z) – (15x + 10y – 8z) =

Change all subtraction within the parentheses first: (8x + –7y + 9z) – (15x + 10y + –8z).

Then, change the subtraction sign outside of the parentheses to addition and change the sign of each term in the polynomial being subtracted: (8x + –7y + 9z) + (–15x + –10y + 8z).

Note that the sign of the term 8z changes twice because it is being subtracted twice.

All that is left to do is combine like terms: 8x + –15x + –7y + –10y + 9z + 8z = –7x + –17y + 17z.

To multiply two polynomials, multiply every term of the first polynomial by every term of the second polynomial. Then, add the products and combine like terms.

Example

(–5x3y)(2x2y3) = (–5)(2)(x3)(x2)(y)(y3) = –10x5y4

To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial and add the products.

Example

6x(10x – 5y + 7) =

Change subtraction to addition: 6x(10x + –5y + 7).

Multiply: (6x)(10x) + (6x)(–5y) + (6x)(7).

Combine like terms: 60x2 + –30xy + 42x.

### 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