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Polynomials are algebraic expressions that consist of variables and coefficients. Variables are sometimes called indeterminates. We can perform arithmetic operations such as addition, subtraction, multiplication, and positive integer exponents for polynomial expressions, but not division by variable. An example of a polynomial with one variable is ${x}^{2}+x-12$ . In this example, there are three terms: ${x}^{2}$ , $x$ , and $-12$ . The single variable is $x$ . This polynomial also contains one exponent, the $2$ in ${x}^{2}$ . Finally, it contains one constant, which is $-12$ .

The word polynomial is derived from the Greek words 'poly' meaning 'many' and 'nominal' meaning 'terms'. So together, it means many terms. A polynomial can have any number of terms, but not an infinite number of terms; we call that a power series. A monomial is one term, such as $3xy$ or ${4}^{2}$ , and polynomials are made up of multiple monomials.

The type of the polynomial is based on how many terms it contains. A binomial contains two terms, such as $3x+{\mathrm{y}}^{2}$ . A trinomial contains three terms, such as $3x+{x}^{2}+14$ . A quadrinomial contains four terms, like $3a\u20134ab+{a}^{2}\u201316$ . The list continues to as many terms as you have in a given problem.

To multiply polynomials, you must use the distributive property.

**Example 1**

Multiply the monomial $3x$ by the binomial $x-2$ .

$3x\left(x\u20132\right)=3x\left(x\right)+3x\left(-2\right)=3{x}^{2}-6x$

When you multiply one binomial by another binomial, you must use the distributive property repeatedly.

**Example 2**

$\left(x+2\right)\left(x-7\right)$

$=x\left(x-7\right)+2\left(x-7\right)$

$=x\left(x\right)+x\left(-7\right)+2\left(x\right)+2\left(-7\right)$

You can use a shortcut to remember how to multiply all of the terms called FOIL, which stands for multiplying the First, Inner, Outer, and Last terms. The product of the two binomials is the sum of four simpler products.

The product of the First terms is $=x\left(x\right)={x}^{2}$

The product of the Outer terms is $=x\left(-7\right)=-7x$

The product of the Inner terms is $=2\left(x\right)=2x$

The product of the Last terms is $=2\left(-7\right)=-14$

Add those up to get the answer:

$={x}^{2}\u20135x\u201314$

You can use a similar strategy to multiply trinomials or other polynomials.

**Example 3**

$\left(x+7y+z\right)\left(a+5\right)$

You need to find six products.

$x\left(a\right)=ax$

$7y\left(a\right)=7ay$

$z\left(a\right)=az$

$x\left(5\right)=5x$

$7y\left(5\right)=35y$

$z\left(5\right)=5z$

Then add the products together to get the answer.

$ax+7ay+az+5x+35y+5z$

College Algebra Diagnostic Tests

Multiplying polynomials takes some practice to get used to. Sometimes the FOIL shortcut is clear and makes it easy to do, and sometimes the shortcut makes it more confusing for students. Either way, if your student is having trouble multiplying polynomials, you should consider having them work with a private tutor. A 1-on-1 tutor can help your student figure out how to multiply polynomials using their most effective learning style so it makes sense to them. They can break the problems down into smaller steps so your student can build up to working out the entire multiplication problem.

By giving your student their full attention, a professional math tutor can take the time needed to make multiplying polynomials come clear to them. To learn how tutoring can help your student multiply polynomials like a champ, contact the Educational Directors at Varsity Tutors today.

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