Algebra Homework Help: How to Multiply Polynomials

Polynomials can often be confusing for algebra students because they are composed of discrete elements that can't simply be combined with one another. To make things even more confusing, many of the elements not only contain variables, but often variables squared and raised to exponents even higher than two. However, it is easy to understand polynomials and how to do basic operations such as multiplying polynomials.

What is a Polynomial Anyway?

A polynomial is a number composed of two or more distinct elements. For example, x + 2 is a polynomial. This is because unless you know the value of x, it cannot be simplified any further. Also, x² + 8x - 3 is a polynomial, this time, with three terms in it.

Polynomials can have any number of terms, and any number of variables, raised to any powers. The only qualification is that each element cannot be combined with another element. For example, 4x + 5x is not a polynomial because it simplifies to 9x, but 4x² + 5x is a polynomial because those elements can't be added.

How do you Multiply Basic Polynomials?

Multiplying two polynomials with each other is a basic operation that involves multiplying each element of one polynomial with each element of the other polynomial. Most basic algebra textbooks start with problems that look like this: (x + 2) (3x - 5)

Students will learn to multiply the first terms, the last terms, the middle terms, and the outside terms to complete the problem. However, this is not the best way to learn to multiply polynomials because with larger polynomials, such as (x + 2) (x² + 3x - 5), the multiplication will not be complete by following just those rules.

Instead, the best way to multiply polynomials is to treat the first polynomial as two separate things, and multiply the second polynomial by each of them. For example, in (x + 2) (3x - 5), treat it as x (3x - 5) plus 2 (3x - 5).

Then to do each of those problems, you would multiply each of the things inside the parentheses by the number outside the parentheses. The answer would be 3x² - 5x + 6x - 10. Then like terms can be combined, so the - 5x and 6x are added and the answer is 3x² + x - 10.

How do you Multiply Bigger Polynomials?

The wonderful thing about this method is that it transfers well to larger polynomials. For example, when faced with the problem (x² + 8x - 3) (2x² - 4x - 1), it can be done with the same method.

Start with the first term in the first polynomial (x²), and multiply it by each term in the second polynomial to get 2x⁴ - 4x³ - x². Then do the same thing, but using 8x to get 16x³ - 32x² - 8x. The last thing to do is to multiply the second polynomial by -3, being careful to pay attention to that negative sign, making -6x² + 12x + 3.

The last step is to simplify the answer of 2x⁴ - 4x³ - x² + 16x³ - 32x² - 8x - 6x² + 12x + 3 by adding like terms, coming up with 2x⁴ + 12x³- 39x² + 4x + 3. This method ensured that none of the 9 multiplications that needed to be done were forgotten, and it can be used on polynomials as big as you want.

How do you Multiply More than two Polynomials?

If you have a problem such as (x + 2) (3x - 5) (2x + 1), the best way to solve it is by multiplying together the first two polynomials, then multiplying the answer (also a polynomial) by the third polynomial.