# PSAT Math: Polynomials

By now you’re used to seeing equations, exponents, and variables; another important topic you are sure to see on the PSAT is polynomials. A **polynomial** is an expression comprised of variables, exponents, and coefficients, and the only operations involved are addition, subtraction, multiplication, division (by constants *only*), and non-negative integer exponents. A polynomial can have one or multiple terms. The following table contains examples of polynomial expressions and non-polynomial expressions.

[raw]

Polynomial | 23x^{ 2} |
x^{5−6} |
y
^{11}−2y^{6}+23xy^{3}−4x^{2} |
47 |

Not a Polynomial |
10z+13 | x^{ 3}y^{ −6} |
x^{12} |
4y − 3 |

[/raw]

### Note

Remember that a constant, such as 47, is considered a polynomial; this is the same as 47*x*^{ 0}. Also, keep in mind that for an expression to be a polynomial, division by a constant is allowed, but division by a variable is not.

Identifying

**like terms**is an important skill that will serve you well on Test Day. To simplify polynomial expressions, you combine like terms just as you did with linear expressions and equations (

*x*terms with

*x*terms, constants with constants). To have like terms, the types of variables present and their exponents must match. For example, 2

*xy*and −4

*xy*are like terms;

*x*and

*y*are present in both, and their corresponding exponents are identical. However, 2

*x*

^{ }

^{2}

*y*and 3

*xy*are not like terms because the exponents on

*x*do not match. A few more examples follow:

Like terms | 7x, 3x, 5x |
3, 15, 900 | xy^{2}, 7xy^{2}, –2xy^{2} |

Not like terms |
3, x, x^{ }^{2} |
4x, 4y, 4z |
xy^{ }^{2}, x^{ }^{2}, 2^{y}xy |

You can also **evaluate** a polynomial expression (just like any other expression) for given values in its domain. For example, suppose you’re given the polynomial expression *x*^{ }^{3} + 5*x*^{ }^{2} + 1. At *x* = –1, the value of the expression is (–1)^{3} + 5(–1)^{2} + 1, which simplifies to –1 + 5 + 1 = 5.

A polynomial can be named based on its **degree**. For a single-variable polynomial, the degree is the highest power on the variable. For example, the degree of 3*x*^{ }^{4} – 2*x*^{ }^{3} + *x*^{ }^{2} – 5*x* + 2 is 4 because the highest power of *x* is 4. For a multi-variable polynomial, the degree is the highest sum of the exponents on any one term. For example, the degree of 3*x*^{ }^{2}*y*^{2} – 5*x*^{ }^{2}*y* + *x*^{ }^{3} is 4 because the sum of the exponents in the term 3*x*^{ }^{2}*y*^{2 }equals 4.

##### Zeros and Roots

On Test Day you might be asked about the nature of the

**zeros**or

**roots**of a polynomial. Simply put, zeros are the

*x*-intercepts of a polynomial’s graph, which can be found by setting each factor of the polynomial equal to 0. For example, in the polynomial equation

*y*= (

*x*+ 6)(

*x*– 2)

^{2}, you would have three equations:

*x*+ 6 = 0,

*x*– 2 = 0, and

*x*– 2 = 0 (because

*x*– 2 is squared, that binomial appears twice in the equation). Solving for

*x*in each yields –6, 2, and 2; we say that the equation has two zeros: –6 and 2. Zeros can have varying levels of

**multiplicity**, which is the number of times that a factor appears in the polynomial equation. In the preceding example,

*x*+ 6 appears once in the equation, so its corresponding zero (–6) is called a

**simple zero**. Because

*x*– 2 appears twice in the equation, its corresponding zero (2) is called a

**double zero**.

You can recognize the multiplicity of a zero from the polynomial’s graph as well. Following is the graph of

*y*= (

*x*+ 6)(

*x*– 2)

^{2}.

When a polynomial has a simple zero (multiplicity 1) or any zero with an odd multiplicity, its graph will cross the

*x*-axis (as it does at

*x*= –6 in the graph above). When a polynomial has a double zero (multiplicity 2) or any zero with an even multiplicity, it just touches the

*x*-axis (as it does at

*x*= 2 in the graph above).

Use your knowledge of polynomials to answer the following test-like question.

### PSAT Math Practice Question: Polynomials

Use the Kaplan Method for Math to solve this question, working through it step-by-step. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.

Strategic Thinking |
Math Scratchwork |

Step 1: Read the question, identifying and organizing important information as you go Don’t let the unusual wording fool you. To find how much greater A is, just do what you would do for two numbers: Subtract the smaller from the larger. |
A – B |

Step 2: Choose the best strategy to answer the question What’s your first step?Substitute the correct expressions for A and B. Distribute the –1 outside the second set of parentheses. Be careful here; this is an easy place to make a mistake.And afterward?Combine like terms. Rearranging so that like terms are next to each other helps here. |
(24xy + 13) – (8xy + 1) = 24 xy + 13 – 8xy – 1=24 xy – 8xy + 13 – 1=16 xy + 12 |

Step 3: Check that you answered the right questionNo further simplification is possible; the correct answer is (C), so you’re done. |