Mid-Chapter Quiz
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Can you use the knowledge of how to multiply rational expressions to multiply a rational expression by a polynomial?
See solution.
We are asked why a student's first step in finding the product 7x* x^3 was to rewrite the expression as shown below. 7/x* x^3=7/x*x^3/1 To answer this question, we will review the definition of a rational expression and how the multiplication of rational expressions works. Let's do it!
Rational Expression:polynomial/polynomial Since any real number can be considered a monomial, which is a type of polynomial, the denominator of a rational expression can be 1. polynomial/1=polynomial We can see that the above rational expression simplifies to be a polynomial. In other words, a polynomial is a rational expression with a denominator of 1. We will use this fact later.
Multiplying rational expressions is equivalent to multiplying their numerators and denominators. Here are some examples.
| Multiplication | Simplify |
|---|---|
| x/x-1*x+1/x^2+2 | x(x+1)/(x-1)(x^2+2) |
| x-1/x^2-4*x^2+6/x-1 | (x-1)(x^2+6)/(x^2-4)(x-1) |
| x-2/x^3+1*x^5+2/x^2-4 | (x-2)(x^5+2)/(x^3+1)(x^2-4) |
We can use the knowledge of how to multiply rational expressions to multiply a rational expression by a polynomial. Earlier we noticed that any polynomial is a rational expression with a denominator of 1. polynomial=polynomial/1 Now we can see that multiplying a rational expression by a polynomial is equivalent to multiplying two rational expressions. Additionally, the Identity Property of Multiplication implies that the denominator will remain the same and we are actually only multiplying the numerator by the polynomial. Again, here are some examples.
| Multiply Rational Expression by Polynomial | Rewrite Polynomial | Simplify |
|---|---|---|
| x/x-1* (x+1) | x/x-1* x+1/1 | x(x+1)/x-1 |
| x^2+2/x+5*(x+5) | x^2+2/x+5*x+5/1 | (x^2+2)(x+5)/x+5 |
| x^2+2/x^2-1* (x-1) | x^2+2/x^2-1* x-1/1 | (x^2+2)(x-1)/x^2-1 |
Considering everything that we reviewed above, we can say that when multiplying a rational expression by a polynomial, a good practice is to write the polynomial as a rational expression. 7/x* x^3=7/x*x^3/1 Rewriting the polynomial as a rational expression helps us to see that we should multiply the numerator by the polynomial and the denominator by 1. 7/x*x^3/1=7x^3/x This is why the student rewrote the multiplication in this way.