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},,,}]

  1. What is the quotient in lowest terms?
  2. 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:

x+8x26x7x2+16x+64x+1\dfrac{\dfrac{x+8}{x^2-6x-7}}{\dfrac{x^2+16x+64}{x+1}}

Step 1: Simplify the expression

To divide by a fraction, we multiply by the reciprocal of the second fraction:

x+8x26x7×x+1x2+16x+64\dfrac{x+8}{x^2-6x-7} \times \dfrac{x+1}{x^2+16x+64}

Now, let's factor the quadratic expressions:

  1. Factor x26x7x^2 - 6x - 7:

    We need two numbers that multiply to 7-7 and add up to 6-6. These numbers are 7-7 and 11. So, we can factor it as:

    x26x7=(x7)(x+1)x^2 - 6x - 7 = (x - 7)(x + 1)

  2. Factor x2+16x+64x^2 + 16x + 64:

    This is a perfect square trinomial, so it factors as:

    x2+16x+64=(x+8)2x^2 + 16x + 64 = (x + 8)^2

Thus, the expression becomes:

x+8(x7)(x+1)×x+1(x+8)2\dfrac{x+8}{(x-7)(x+1)} \times \dfrac{x+1}{(x+8)^2}

Step 2: Cancel out common factors

We can cancel out the terms that appear in both the numerator and denominator:

  • (x+8)(x + 8) appears in the numerator of the first fraction and the denominator of the second fraction.
  • (x+1)(x + 1) appears in the denominator of the first fraction and the numerator of the second fraction.

After canceling, we are left with:

1(x7)(x+8)\dfrac{1}{(x-7)(x+8)}

This is the quotient in its lowest terms.

Step 3: Determine the excluded values

We must exclude values of xx that make any denominator equal to zero in the original expression.

  • From x26x7=(x7)(x+1)x^2 - 6x - 7 = (x-7)(x+1), we get the potential values x=7x = 7 and x=1x = -1.
  • From x2+16x+64=(x+8)2x^2 + 16x + 64 = (x+8)^2, we get the potential value x=8x = -8.

Therefore, the values of xx that must be excluded from the domain are x=8x = -8, x=1x = -1, and x=7x = 7.

Final Answer:

  1. The quotient in lowest terms is:

    1(x7)(x+8)\dfrac{1}{(x - 7)(x + 8)}

  2. The values of xx that must be excluded are:

    • Choice A: x=8x = -8
    • Choice B: x=1x = -1
    • Choice D: x=7x = 7

Would you like further details on how to approach similar problems or additional explanations?

Related questions:

  1. How do you simplify complex rational expressions involving multiple terms?
  2. What is the domain of a rational expression, and how do we find it?
  3. How can factoring help in simplifying rational expressions?
  4. What is the process of multiplying rational expressions, and how is it different from division?
  5. 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