Simplifying and Evaluating Polynomials with More Than One Term

Learning Objectives

• Evaluate a polynomial for given values of each variable.
• Simplify polynomials by collecting like terms.

You have studied polynomialsA monomial or the sum or difference of two or more monomials. 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 `5x^4 + 3x^3 - 6x^2 + 2x` containing one variable, but polynomials can also contain multiple variables. An example of a polynomial with two variables is `4x^2y - 2xy^2 + 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 review 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.

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: `t^3 - 10t^2 - 5t - 32`. This polynomial has been written in descending order of degree, starting with the term A number or product of a number and variables raised to powers. `4x, -5y^2, 6`, and `x^3y^4` are all examples of terms. 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 monomialA polynomial with exactly one term. `4x`, `-5y^2`, and `6` are all examples of monomials. `t^3`. (Note that the degree of a monomialThe degree of a monomial is the power to which the variable is raised. For example, the monomial `5y^2` has a degree of `2`. If the monomial contains several variables then the degree of the monomial is the sum of the degree of all the variables. For example, the monomial `7x^2y^3` has a degree of `5`. , `t^3`, 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: `7x^2y - 3xy^3 + 2x`. This polynomial has three terms and therefore can be called a trinomial A polynomial with exactly three terms, such as `5y^2 - 4y + 4` and `x^2 + 2xy +y^2`. . To determine the degree of a term, you find the sum of the exponents of all the variables in the term. Here are some examples:

The degree of a polynomialThe highest exponent or sum of exponents of a term in a polynomial. For example, `7x^2y^3 + 3x^2y - 8` is a `5`th degree polynomial because the highest sum of exponents in a term is `2 + 3 = 5`. is the same as the degree of the term with the highest degree. In this case, `7x^2y - 3xy^3 + 2x` is a fourth-degree polynomial.

If a polynomial has like termsTerms that contain the same variables raised to the same powers. For example, `3x` and `−8x` are like terms, as are `8xy^2` and `0.5xy^2`., the polynomial can be simplified by combining the like terms.

You’ll recall that like terms contain the same variables raised to the same 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 `3xy^3z^2 + 5xy^3z^2 + 6x^2y^3z` has like terms that can be combined. `3xy^3z^2` and `5xy^3z^2` are like terms because they have the same variables, `x`, `y`, and `z`, raised to the same powers, `x`, `y^3`, and `z^2`. They can be collected, or combined, to give a result of `8xy^3z^2`. Notice that while `6x^2y^3z` has the same variables, `x`, `y`, and `z`, the exponents in this term are different, `x^2` instead of `x`, and `z` instead of `z^2`. So, `6x^2y^3z` cannot be combined with the other terms. Instead, the simplified polynomial is written with two terms: `8xy^3z^2 + 6x^2y^3z`.

As 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.

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.