Math Problem Statement
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Level 1 0 /1 skill Course: Precalculus
Unit 4 Lesson 6: Multiplying and dividing rational expressions Math>Precalculus>Rational functions>Multiplying and dividing rational expressions © 2025 Khan Academy Terms of usePrivacy PolicyCookie Notice
Multiply & divide rational expressions Google Classroom
Microsoft Teams Problem Answer two questions about the following rational division.
[\dfrac{\dfrac{x+8}{x^2-6x-7}}{,,,\dfrac{x^2+16x+64}{x+1},,,}]
- What is the quotient in lowest terms?
- What values of
[x] must we exclude from the domains of the expressions?
Choose all answers that apply:
Choose all answers that apply:
(Choice A)
[x=-8] A [x=-8] (Choice B)
[x=-1] B [x=-1] (Choice C)
[x=1] C [x=1] (Choice D)
[x=7] D [x=7] Related content Report a problem
Do 4 problems You completed your assignment for PD3 - IM3.
Solution
Let's tackle the given rational expression step by step.
The problem involves dividing two rational expressions:
Step 1: Simplify the expression
To divide by a fraction, we multiply by the reciprocal of the second fraction:
Now, let's factor the quadratic expressions:
-
Factor :
We need two numbers that multiply to and add up to . These numbers are and . So, we can factor it as:
-
Factor :
This is a perfect square trinomial, so it factors as:
Thus, the expression becomes:
Step 2: Cancel out common factors
We can cancel out the terms that appear in both the numerator and denominator:
- appears in the numerator of the first fraction and the denominator of the second fraction.
- appears in the denominator of the first fraction and the numerator of the second fraction.
After canceling, we are left with:
This is the quotient in its lowest terms.
Step 3: Determine the excluded values
We must exclude values of that make any denominator equal to zero in the original expression.
- From , we get the potential values and .
- From , we get the potential value .
Therefore, the values of that must be excluded from the domain are , , and .
Final Answer:
-
The quotient in lowest terms is:
-
The values of that must be excluded are:
- Choice A:
- Choice B:
- Choice D:
Would you like further details on how to approach similar problems or additional explanations?
Related questions:
- How do you simplify complex rational expressions involving multiple terms?
- What is the domain of a rational expression, and how do we find it?
- How can factoring help in simplifying rational expressions?
- What is the process of multiplying rational expressions, and how is it different from division?
- How do we identify and exclude values from the domain in rational functions?
Tip: Always check for factors that can be canceled out before simplifying to make your work easier.
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Math Problem Analysis
Mathematical Concepts
Rational Expressions
Algebra
Factoring
Domain of Rational Functions
Formulas
Division of Rational Expressions
Factorization of Quadratic Equations
Theorems
Reciprocal of a Fraction
Domain of Rational Functions
Suitable Grade Level
Grades 9-12
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