Problem 33 - Elementary Algebra

Question

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 \(\frac{x+2}{x-9}\).

Step-by-Step Solution

Answer

The numerator is x + 2.

Step 1: 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: \[x^2 - 12x + 27 = (x - 3)(x - 9)\].

Step 2: Factor the Numerator of the Second Fraction

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

Step 3: Set Up the Given Product

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

Step 4: Simplify the Expression

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

Step 5: 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:\[A = x + 2\].

Key Concepts

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.