Simplifying and Evaluating Polynomials with More Than One Term

 

Learning Objective(s)

·         Evaluate a polynomial for given values of each variable.

·         Simplify polynomials by collecting like terms.

 

 

You have studied polynomials consisting of constants and/or variables combined by addition or subtraction. The variables may include exponents. The examples so far have been limited to expressions such as 5x⁴ + 3x³ – 6x² + 2x containing one variable, but polynomials can also contain multiple variables. An example of a polynomial with two variables is 4x²y – 2xy² + x – 7.

 

Many formulas are polynomials with more than one variable, such as the formula for the surface area of a rectangular prism: 2ab + 2bc + 2ac, where a, b, and c are the lengths of the three sides. By substituting in the values of the lengths, you can determine the value of the surface area. By applying the same principles for polynomials with one variable, you can evaluate or combine like terms in polynomials with more than one variable.

 

 

When you evaluate an expression for a given value, you substitute that given value in the expression, and find its numerical value. In the following example, x = −2, you replace all of the x’s with a value of −2 and simplify the expression following the order of operations.

 

 

 

 

You can follow the same procedure when there are two variables in an expression. Let’s look at an example.

 

 

 

 

As with polynomials with one variable, you must pay attention to the rules of exponents and the order of operations so that you correctly evaluate an expression with two or more variables.

 

 

 

 

 

 

The next example shows how to evaluate a polynomial with two variables. This polynomial is the formula for perimeter of a rectangle.

 

 

 

 

Evaluate 2x3 – xy2 + 6 for x = −2 and y = 5   A) −158   B) −60   C) 14   D) 40     Show/Hide Answer A) −158 Incorrect. Unless there are parentheses, an exponent only goes with the variable immediately to its left. So, the exponent 3 does not apply to the 2 in that term, only to the variable x. This is true in the next term also, x is not squared. The correct answer is 40.   B) −60 Incorrect. Be more careful with your sign work, the middle term is positive 50, not negative 50.When you substitute in the values of the variables, you get 2(−2)3 – (−2)(5)2 + 6, which is simplified to 2(−8) – (−2)(25) + 6, which equals −16 + 50 + 6. The correct answer is 40.   C) 14 Incorrect. An exponent indicates the number of times a value is used as a factor. x3 means  “x times x times x”, not “x times 3.”  2(−2)( −2)( −2) – (−2)(5)(5) + 6  = −16 + 50 + 6. The correct answer is 40.   D) 40 Correct. 2(−2)3 – (−2)(5)2 + 6 = −16 + 50 + 6 = 34 + 6 = 40.

 

 

 

Mathematicians use conventions for writing and describing polynomials. A polynomial with one variable can be described by the number of terms it has and the degree of the term with the greatest exponent. Polynomials are commonly written with their terms in descending order of degree. Let’s start by looking at an example of a polynomial with one variable: t³ – 10t² – 5t – 32. This polynomial has been written in descending order of degree, starting with the term with an exponent of 3 and ending with the term whose degree is 0 because it has no variable. This polynomial is called a third degree polynomial because its term with the highest degree is the monomial t³. (Note that the degree of a monomial, t³, is also 3, because the variable t has an exponent of 3.)

 

When a polynomial has more than one variable, you can still describe it according to its degree and the degree of its terms. It’s a little more complicated. Let’s look at a polynomial with two variables: 7x²y – 3xy³ + 2x. This polynomial has three terms and therefore can be called a trinomial. To determine the degree of a term, you find the sum of the exponents of all the variables in the term.

 

 

The degree of a polynomial is the same as the degree of the term with the highest degree. In this case, 7x²y – 3xy³ + 2x is a fourth degree polynomial.

 

 

What description below best matches the expression: 2x4y – 5x3 – 10xy3?   A) A twelfth degree trinomial   B) A fifth degree trinomial   C) A third degree polynomial   D) A fourth degree polynomial   Show/Hide Answer A) A twelfth degree trinomial Incorrect. The degree of a polynomial is the same as the degree of the term whose exponents have the greatest sum, not by the sum of all the exponents in the polynomial. The correct answer is a fifth degree trinomial.   B) A fifth degree trinomial Correct. 2x4y has a degree of 5 because the exponents of its variables are 4 and 1, and 4 + 1 = 5. This polynomial is called a trinomial because it has 3 terms.   C) A third degree polynomial Incorrect. The degree of a polynomial is not determined by the number of terms it has. The degree is the same as the degree of the term whose exponents have the greatest sum. The correct answer is a fifth degree trinomial.   D) A fourth degree polynomial Incorrect. When there are multiple variables, find the degree of a polynomial by determining which term’s exponents have the greatest sum. This is the degree of the polynomial. The 2x4y term is the term with the highest degree, the sum of the exponents in the term 2x4y is 4 + 1 = 5. Since there are 3 terms, this polynomial is called a trinomial. The correct answer is a fifth degree trinomial.

 

 

 

If a polynomial has like terms the polynomial can be simplified by combining the like terms.

 

You’ll recall that like terms contain the same exact variables raised to the same exact power. If there is more than one variable, the same is true – same exact variable(s) each raised to the same exact power.

 

The polynomial 3xy³z² + 5xy³z² + 6x²y³z has like terms that can be combined. 3xy³z² and 5xy³z² are like terms because they have the same exact variables, x, y, and z, raised to the same exact powers, x, y³, and z². They can be collected, or combined, to give a result of 8xy³z². Notice that while 6x²y³z has the same variables, x, y, and z, the exponents in this term are different, x² instead of x, and z instead of z². So, 6x²y³z cannot be combined with the other terms. Instead, the simplified polynomial is written with two terms: 8xy³z²  + 6x²y³z.

 

 

 

Like with polynomials with one variable, you can combine like terms in polynomials with more than one variable by combining the coefficients of those like terms and keeping the variable part the same. That step is written in the example below. But to save time, you can also just perform the computation in your head.

 

 

 

 

Simplify by collecting like terms: 4(x2y  + 7y) – 5y(3x2 – y) – 10y   A) −11x2y + 5y2 + 18y   B) 4x2y + 11y – 8yx2 – 16y   C) 4x2y + 18y – 15yx2 + 5y2   D) 4x2y – 5y(3x2 – y) – 3y     Show/Hide Answer A) −11x2y + 5y2 + 18y Correct. 4(x2y  + 7y) – 5y(3x2 – y) – 10y = 4x2y + 28y – 15yx2 + 5y2 – 10y = 4x2y + 18y – 15yx2 + 5y2 = −11x2y + 5y2 + 18y.   B) 4x2y + 11y – 8yx2 – 16y Incorrect. The 4 is distributed to both terms in the parentheses by multiplying each term by 4 resulting in 28y, not 11y. The −5y is similarly distributed to each term in the parentheses, resulting in −15yx2 + 5y2. The correct answer is −11x2y + 5y2 + 18y.   C) 4x2y + 18y – 15yx2 + 5y2 Incorrect. This polynomial can be further simplified by combining the like terms 4x2y and −15yx2. The order of the variables in a term does not matter. The correct answer is −11x2y + 5y2 + 18y.   D) 4x2y – 5y(3x2 – y) – 3y Incorrect. Before combining like terms, you must distribute to clear the parentheses. The correct answer is −11x2y + 5y2 + 18y.

 

 

 

Polynomials can contain more than one variable and can be evaluated in the same way as polynomials with one variable. To evaluate any polynomial, you substitute the given values for the variable and perform the computation to simplify the polynomial to a numerical value. The order of operations and integer operations must be properly applied to correctly evaluate a polynomial.

 

Polynomials with more than one variable can be simplified by combining like terms, as you can do with polynomials with one variable. Like terms must contain the same exact variables raised to the same exact power. In terms with more than one variable, the order in which the variables are written does not matter.