Recall that we can use the equivalent fractions property to simplify fractions by dividing out common factors from the numerator and the denominator. In this section, you will learn how to divide a polynomial by a monomial using the equivalent fractions property and the quotient property of exponents.
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Example 1: Divide the expression with a numerator of 42 x squared times y cubed and a denominator of negative 7 x times y to the power of 5.
Line 1: Using fraction multiplication, write one fraction for each numerical coefficient and for each variable, so the expression is 42 over negative 7 times x squared over x times y cubed over y to the power of 5.
Line 2: Simplify each fraction using the quotient property of exponents so the expression is negative 6 times x times 1 over y squared.
Line 3: Multiply to get the final simplified expression with a numerator of negative 6x and a denominator of y squared.
Example 2. Find the quotient of the expression with a numerator of 3x cubed y squared times 10x squared y cubed and a denominator of 6 x to the power of 4 times y to the power of 5.
Line 1: The fraction bar is a grouping symbol, so the numerator must be simplified first. Simplify the numerator by using the product property of exponents so the expression has a numerator of 30 x to the power of 5 times y to the power of 5 and the denominator is 6 x times y to the power of 5.
Line 2: Use fraction multiplication to write one fraction for the numerical coefficient and for each variable so the expression is 30 over 6 times x to the power of 5 over x to the power of 4 times y to the power of 5 over y to the power of 5.
Line 3: Simplify each fraction using the quotient property of exponents so the expression is 5 times x times 1.
Line 4: Multiply so the final simplified expression is 5x.
Example 3: Simplify the expression with a numerator of 2 x cubed y squared plus 10x squared y cubed minus 6x y squared and a denominator of 4 x y cubed.
Line 1: Determine the greatest common factor between each term in the numerator and the denominator is 2 x y squared.
Line 2: Factor out 2 x y squared from each term in the numerator and the denominator, so the expression can be written with a numerator of 2 x y squared times open parenthesis x squared plus 5 x y minus 3 close parentheses, and a denominator of 2 x y squared times 2y.
Line 3: Remove the common factors in the numerator and denominator to simplify, so the final expression has a numerator of x squared plus 5 x y minus 3 and a denominator of 2y.
Examples Source: "Prealgebra - opens in a new window" by Lynn Marecek & Mary Anne Anthony-Smith is licensed under CC BY 4.0 - opens in a new window / A derivative from the original work - opens in a new window