Problem 22 - Elementary and Intermediate Algebra

Question

Perform each multiplication. a. \(4 x\left(\frac{3}{4 x}\right)\) b. \((x+6)(x-2)\left(\frac{3}{x-2}\right)\) c. \(8(x+4)\left[\frac{7 x}{2(x+4)}\right]\) d. \(6(m-5)\left(\frac{7}{5-m}\right)\)

Step-by-Step Solution

Answer

a. 3; b. 3x + 18; c. 28x; d. -42.

Step 1: Simplify and Multiply Expression (a)

The expression is \( 4x \left( \frac{3}{4x} \right) \). Notice that \( 4x \) and \( \frac{1}{4x} \) are reciprocal terms. Thus, they will cancel each other out when multiplied together. Therefore, the expression simplifies to:\[4x \left( \frac{3}{4x} \right) = 3\]The final result for expression (a) is 3.

Step 2: Simplify and Multiply Expression (b)

The original expression is \( (x+6)(x-2)\left(\frac{3}{x-2}\right) \). First, notice that \( (x-2) \) is in both the numerator and denominator; they cancel each other out. Thus, the expression simplifies to:\[(x+6) \cdot 3\]Now, distribute the 3:\[(x+6) \cdot 3 = 3x + 18\]The final result for expression (b) is \( 3x + 18 \).

Step 3: Simplify and Multiply Expression (c)

The given expression is \( 8(x+4)\left(\frac{7x}{2(x+4)}\right) \). Notice \( (x+4) \) in the numerator and denominator can cancel each other out. Simplifying, we get:\[8 \cdot \frac{7x}{2} = 8 \cdot \frac{7x}{2}\]Cancel out the common factor of 2:\[= 4 \cdot 7x = 28x\]The final result for expression (c) is \( 28x \).

Step 4: Simplify and Multiply Expression (d)

The expression is \( 6(m-5)\left(\frac{7}{5-m}\right) \). Recognize that \( m-5 \) and \( 5-m \) are negatives of each other: \[-1(5-m) = (m-5)\]Thus, their product is -1:\[= 6 \cdot \frac{-7}{(m-5)}\]This becomes:\[-6 \cdot 7 = -42 \]Therefore, the final result for expression (d) is -42.

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial part of solving algebraic problems. When we simplify, we aim to make expressions as straightforward and uncomplicated as possible. This involves reducing fractions, canceling out unnecessary terms, and combining like terms.
For example, in the expression \( 4x \left( \frac{3}{4x} \right) \), the term \( 4x \) in the numerator and the denominator cancels out, which leaves us with a simple numerical outcome of 3.
Simplification makes expressions easier to work with and solve. Keeping an eye out for common factors and equivalent terms can help in this process. Overall, simplifying expressions is a vital skill that can save time and reduce error risk in calculations.

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, composed under operations of addition, subtraction, and multiplications by non-negative integer exponents. They are the building blocks of many algebraic equations.
In the context of the problem, expression (b) involved polynomials: \((x+6)\) and \((x-2)\). When multiplying these, first simplify by canceling out similar terms, like the \((x-2)\) which appears in both a numerator and denominator.
Once simplified, distribute any remaining terms to get linear expressions. For example, distributing 3 to each part of \((x+6)\) gives us \(3x + 18\).
Understanding polynomials is important as they frequently appear in algebra, and mastering their manipulation is crucial for more complex mathematical problems.

Rational Expressions

Rational expressions are fractions where the numerator, the denominator, or both, are polynomials. Simplifying rational expressions often involves canceling similar terms in both the numerator and the denominator to reduce the expression to its simplest form.

• In example (c): The expression \(8(x+4)\left(\frac{7x}{2(x+4)}\right)\) simplifies when \((x+4)\) terms cancel, resulting in a simpler fraction. Simplifying further, by handling coefficients and like terms, we arrive at a straightforward expression, \(28x\).
• In example (d): Recognizing that \((m-5)\) and \((5-m)\) resemble each other but differ by a negative sign helps simplify the expression, ultimately giving a result of -42 after addressing the negative product.

Rational expressions require a solid understanding of factoring and simplifying fractions. These skills are essential to ensure clear and correct solutions in algebra.