Adding and Subtracting Polynomials Symbolically

When adding and subtracting polynomials, terms with common variables (with the same exponent(s)) can be combined by adding or subtracting the coefficients. Terms with common variables are often called like terms. Adding and subtracting these like terms to simplify an expression is often called 'combining like terms'.

Like Terms
terms with the same variable(s) (with the same exponent(s))

Example 7

  1. If possible, simplify each of the following expressions.

    1.   «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«mi»x«/mi»«/math»

      This expression can be simplified because the terms are like terms. Add the coefficients to determine the sum.



    2.   «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»5«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»4«/mn»«mi»x«/mi»«/math»

      This expression cannot be simplified because the two terms are not like terms. Although they both involve the variable x, the exponents differ.

    3.   «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»7«/mn»«mi»x«/mi»«mi»y«/mi»«mo»-«/mo»«mn»9«/mn»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»5«/mn»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mi»x«/mi»«mi»y«/mi»«/math»

      This expression contains two pairs of like terms, so it can be simplified. Begin by rearranging the polynomial so like terms are beside each other.