Problem 37 - College Algebra and Calculus: An Applied Approach

Question

Perform the indicated operations and simplify.\(\frac{(x-9)(x+7)}{x+1} \cdot \frac{x}{9-x}\)

Step-by-Step Solution

Answer

-(x(x+7))/(x+1) or -(x^2+7x)/(x+1)

Step 1: Distribute the expressions in the numerator and denominator of the fractions

First, distribute the expressions in the numerator and denominator of each fraction, simplifying the expressions: \((x-9)(x+7)\) in the numerator becomes \(x^2 -2x -63\), and \(x+1\) in the denominator stays as it is. For the second fraction, the numerator \(x\) stays as it is, and \(9-x\) in the denominator becomes \(-x + 9\). So our expression now looks like this: \(\frac{x^2-2x-63}{x+1} \cdot \frac{x}{-x+9}\).

Step 2: Factor the quadratic expression

Next, factor the quadratic expression \(x^2 -2x -63\) in the first fraction. The factors are \((x-9)(x+7)\). Now our expression looks like this: \(\frac{(x-9)(x+7)}{x+1} \cdot \frac{x}{-x+9}\).

Step 3: Simplify the expression

Now, rewrite \(-x+9\) as \(-(x-9)\) for easier cancellation in the next step. Now our expression is: \(\frac{(x-9)(x+7)}{x+1} \cdot \frac{x}{-(x-9)}\).

Step 4: Cancel out like terms

Now, notice that \((x-9)\) terms can be cancelled out, leaving us with: \(\frac{(x+7)}{x+1} \cdot (\frac{x}{-1})\), which simplifies to \(-\frac{(x+7)x}{x+1}\)

Step 5: Simplify Further

You can leave the expression as \(-\frac{(x+7)x}{x+1}\) or distribute \(x\) in the numerator to get \(-\frac{x^2+7x}{x+1}\). Either of them is a correct simplified form.

Key Concepts

Polynomial Multiplication

When multiplying polynomials, each term in the first polynomial is multiplied by each term in the second polynomial. This method is often called the distributive property or using FOIL (First, Outer, Inner, Last) for binomials. For example, given the polynomials

• \((x - 9)(x + 7)\)

we distribute or FOIL to find:

• First: \(x \times x = x^2\)
• Outer: \(x \times 7 = 7x\)
• Inner: \(-9 \times x = -9x\)
• Last: \(-9 \times 7 = -63\)

Combine these to get the expanded form:

• \(x^2 - 2x - 63\)

This process sets up the terms for factoring back into a product of simpler binomials, if possible.

Quadratic Factoring

Factoring quadratics splits a polynomial into products of smaller polynomials. Consider the quadratic

• \(x^2 - 2x - 63\)

The objective is to express this as

• \((x - a)(x - b)\)

where \(a\) and \(b\) are numbers that multiply to \(-63\) and add to \(-2\).
Here,

• \(-9 \times 7 = -63\) and \(-9 + 7 = -2\)

Thus, the factors are:

• \((x - 9)(x + 7)\)

Factoring reduces complexity and allows for simplification, critical for solving equations and simplification in algebraic fractions.

Fraction Simplification

Simplification of algebraic fractions involves reducing expressions by canceling common factors from the numerators and denominators. Consider the original expression:

• \(\frac{(x - 9)(x + 7)}{x + 1} \cdot \frac{x}{-(x - 9)}\)

Notice the

• \((x - 9)\)

in both the numerator and denominator. Cancel these common factors:

• \(\frac{(x + 7)}{x + 1} \cdot \frac{x}{-1}\)

This further reduces to:

• \(-\frac{(x+7)x}{x+1}\)

Understanding how to simplify involves recognizing factors and using them to make expressions more manageable, essential for evaluating and solving polynomial equations.