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Chapter 7: Problem 33

Fill in the numerator of \(\frac{?}{x^{2}+4 x-32} \cdot \frac{x^{2}+5 x-24}{x^{2}-12 x+27}\) so that the product is \(\frac{x+2}{x-9}\).

Short Answer

Expert verified
The numerator is x + 2.

Step by step solution

01

Factor the Denominators

First, factor both denominators. For the first fraction, factor the quadratic polynomial in the denominator: \[x^2 + 4x - 32 = (x + 8)(x - 4)\]. For the second fraction: \[x^2 - 12x + 27 = (x - 3)(x - 9)\].
02

Factor the Numerator of the Second Fraction

Factor the numerator of the second fraction: \[x^2 + 5x - 24 = (x + 8)(x - 3)\].
03

Set Up the Given Product

Express the given product using the factors from the previous steps and the unknown numerator \((A)\).\[\frac{A}{(x + 8)(x - 4)} \cdot \frac{(x + 8)(x - 3)}{(x - 3)(x - 9)} = \frac{x + 2}{x - 9}\].
04

Simplify the Expression

Simplify the expression by canceling out common factors. The equation simplifies to:\[\frac{A}{(x - 4)} \cdot \frac{1}{(x - 9)} = \frac{x + 2}{x - 9}\].
05

Determine the Numerator

Since the right-hand side of the equation requires the denominator on the left to be \((x - 9)\), set the numerators equal:\[A = x + 2\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

quadratic polynomial
In this exercise, we encounter quadratic polynomials while factorizing the denominators and numerators. A quadratic polynomial takes the general form of \( ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants. To factorize these polynomials, we need to find two binomials that multiply together to give the original quadratic polynomial.
  • Consider the polynomial \( x^2 + 4x - 32 \).
    Think about which two numbers multiply to \(-32\) and add up to \4\. These numbers are \8\ and \-4\.
    So, we can factor it as \( (x + 8)(x - 4) \).
  • For the polynomial \( x^2 - 12x + 27 \), we need numbers that multiply to \27\ and add up to \-12\. These numbers are \(-3\) and \(-9\).
    Therefore, the factor form is \( (x - 3)(x - 9) \).
Mastering the art of recognizing and factorizing quadratic polynomials is crucial, so practice this skill often.
denominator
The denominator of a fraction represents the number of equal parts the whole is divided into. In this exercise, we focus on simplifying and managing the denominators of compounded fractions.
Denominators can sometimes be quadratic polynomials, as seen here. To simplify the fraction, we factored the denominators:
  • First fraction denominator: \( x^2 + 4x - 32 \) became \( (x + 8)(x - 4) \).
  • Second fraction denominator: \( x^2 - 12x + 27 \) became \( (x - 3)(x - 9) \).
Factoring the denominators allows for term cancellation across the fractions, making the overall expression simpler. Always look out for common factors in the denominators and numerators to streamline your solution.
simplification
Simplification plays a pivotal role in solving the given exercise. We simplify expressions to transform them into more manageable forms without changing their values.
In our problem, we simplified by canceling common factors:
\( \frac{A}{(x-4)} \times \frac{1}{(x-9)} = \frac{x+2}{(x-9)} \).
The step-by-step process involved:
  • Initially factoring the numerator and denominators fully.
  • Canceling out common factors across the numerators and denominators.
  • Focusing on matching the simplified expression on both sides of the equation.
By methodically simplifying complex fractions and ensuring the expressions match, we found: \( A = x + 2 \).
Remember, the goal of simplification is to make the problem easier to solve while keeping the mathematical equivalence intact.

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Most popular questions from this chapter

For exercises \(67-82\), use the five steps and a proportion. In 2010 , there were \(14.9\) cases of syphilis per 100,000 Americans with a total of 45,834 cases of syphilis. Find the population of Americans used to create this ratio. Round to the nearest hundred. (Source: www.cdc.gov, 2011)

For exercises 1-10, (a) solve. (b) check. $$ \frac{2}{3} x+\frac{3}{2}=\frac{1}{3} x+\frac{1}{6} $$

For exercises 43-58, (a) solve. (b) check. $$ \frac{z+2}{4}=\frac{z-8}{12} $$

For exercises \(67-82\), use the five steps and a proportion. Find the number of adults used to create the ratio "four out of five." Four out of five adults now use the Internet. 184 million adults are online from their homes, offices, schools or other locations. (Source: www.harrisinteractive.com, Nov. 17, 2008)

When the top of a cone is removed, the formula for the volume of the remaining cone (the frustrum) is \(V=\frac{1}{3} \pi\left(R^{2}+R r+r^{2}\right) h\), where \(r\) is the radius of the circle at the top of the frustrum and \(R\) is the radius of the circle at the bottom of the frustrum. In 1856, an American army officer, Henry Hopkins Sibley, invented and received a patent for the design of a conical tent that could sleep 12 soldiers. (The apex is the diameter of the top of the frustrum.) Find the volume of the tent in cubic feet. Use \(\pi \approx 3.14\). Round to the nearest whole number. Be it known that I, H.H. Sibley, United States Army, have invented a new and improved Conical Tent ... the tent is in shape the frustrum of a cone; the base 18 feet; the height 12 feet; the apex 1 foot 6 inches [1.5 ft]. (Source: patimg1.uspto.gov)
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