Write each rational expression in lowest terms. See Example 2....

Question

Write each rational expression in lowest terms. See Example 2. (m² - 4m + 4) / (m² + m - 6)

Accepted Answer

Hey, everyone here, we are asked to reduce the following rational expression to the lowest terms and indicate the values that should not be included in its domain. Here, we are given the expression Z squared minus 12 Z plus 36 all divided by Z squared plus Z minus 42. Here we have four answer choice options, answer choices A through D, each of which vary in the reduced rational expression as well as the values that must be excluded from the domain. So to begin reducing our given expression, we first want to factor the numerator and the denominator to see if there are any expressions that we can simplify. So starting in the numerator, we can factor its trinomial expression into the quantity of Z minus six multiplied by the quantity of Z minus six. And in the denominator, we can factor its expression into the quantity of Z minus six multiplied by the quantity of Z plus seven. So now that we have factored our given expression, we actually want to determine the values that cannot be included in the domain by looking at our denominator. So looking at the expression in the denominator, we have the quantity of Z minus six multiplied by the quantity of Z plus seven. And we know that this expression cannot be equal to zero. Since if it is zero, this will cause us to divide by zero. And thus cause our expression to be undefined. So we want to take our denominator and set it equal to zero and solve for the Z values that cause the expression to be equal to zero Since those will be the values that must be excluded from the domain. So to find these Z values, we want to recall the zero product property which allows us to set each individual expression equal to zero and solve for Z. So starting with the first part of our expression, we have Z minus six. Again, we set this expression equal to zero and solve for Z. So just adding six to both sides, we find Z is equal to six. And now repeating with the other part of our expression, we have Z plus seven is equal to zero solving for Z, we subtract seven from both sides. So we find Z is equal to negative seven. And now we know that Z cannot be equal to six and Z cannot be equal to negative seven since these values cause our expression to be undefined and therefore must be excluded from the domain. So now looking back at our factored expression, recalling in the numerator, we have the quantity of Z minus six multiplied by the quantity of Z minus six. And in the denominator, we have the quantity of Z minus six multiplied by the quantity of Z plus seven. Now, we want to start reducing this expression by noticing we have Z minus six in both the numerator and the denominator. So both expressions cancel and we can now rewrite our expression in its reduced form as Z minus six all divided by Z plus seven. And now we see that we cannot reduce our expression any further. So this is our final simplified expression along with the values that must be excluded from the domain. And so we are left with answer choice. A where again, our reduced expression is the quantity of Z minus six divided by the quantity of Z plus seven where Z cannot be equal to negative seven or positive six. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Write each rational expression in lowest terms. See Example 2. (m² - 4m + 4) / (m² + m - 6)

Key Concepts

Factoring Quadratic Expressions

Simplifying Rational Expressions

Identifying and Excluding Restrictions

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