Problem 91 - Intermediate Algebra

Question

Look Alikes . . . a. \(\frac{t^{2}+9 t+20}{9 t+36} \cdot \frac{9 t+45}{t+4}\) b. \(\frac{t^{2}+9 t+20}{9 t+36} \div \frac{9 t+45}{t+4}\)

Step-by-Step Solution

Answer

a: \((t+5)^2\); b: 1

Step 1: Factor Numerators and Denominators

First, factor each expression in the numerators and denominators given in the problem.For (a) and (b), we have:1. **Numerator** of the first fraction: \( t^{2} + 9t + 20 \) factors to \((t+5)(t+4)\).2. **Denominator** of the first fraction: \( 9t + 36 \) factors to \(9(t+4)\).3. **Numerator** of the second fraction in both (a & b): \( 9t + 45 \) factors to \(9(t+5)\).4. **Denominator** of the second fraction in both (a & b): \( t + 4 \) does not factor further.

Step 2: Simplify Multiplication Expression (Part a)

For part (a), The expression becomes:\[ \left( \frac{(t+5)(t+4)}{9(t+4)} \right) \cdot \left( \frac{9(t+5)}{t+4} \right) \]Cancel out the common terms:- \((t+4)\) cancels from the first fraction and numerator of the second fraction.- 9 cancels with 9 from both fractions.The simplified expression is:\[ (t+5)(t+5) = (t+5)^2 \]

Step 3: Simplify Division Expression (Part b)

For part (b), we need to divide the first fraction by the second:The expression becomes:\[ \frac{(t+5)(t+4)}{9(t+4)} \div \frac{9(t+5)}{t+4} \]Instead of division, rewrite as multiplication by flipping the second fraction:\[ \left( \frac{(t+5)(t+4)}{9(t+4)} \right) \cdot \left( \frac{t+4}{9(t+5)} \right) \]Cancel out the common terms:- \((t+5)\) cancels from both fractions.- \((t+4)\) cancels as well.- 9's can be canceled as they appear in both fractions.The simplified expression is:\[ 1 \]

Step 4: Conclusion and Simplified Results

1. For part (a), after simplification, the final expression is \((t+5)^2\).2. For part (b), the final expression simplifies to 1.

Key Concepts

Rational Expressions

Rational expressions are fractions with polynomials in their numerators and denominators. It's similar to the way regular fractions work, but instead of numbers, we deal with algebraic expressions. The core idea is that whatever is in the bottom (denominator) of the fraction cannot be zero, just like you can't divide by zero in basic arithmetic.

• Example: In the expression \( \frac{t^2 + 9t + 20}{9t + 36} \), the numerator is \( t^2 + 9t + 20 \), and the denominator is \( 9t + 36 \).
• Key Point: If the denominator ever becomes zero, the expression becomes undefined.

When dealing with rational expressions, it's crucial to always consider the values that make the denominator zero, as they must be excluded from the solution set.

Factoring Polynomials

Factoring is breaking down a polynomial into simpler 'factor' forms, much like splitting 12 into 3 and 4. For polynomials, this involves expressing it as a product of two or more polynomials. Factoring reveals common factors and helps simplify expressions.Consider the expression \( t^2 + 9t + 20 \). Factoring involves finding two numbers that multiply to 20 and add to 9. These numbers are 4 and 5, giving us:

• \( t^2 + 9t + 20 = (t+4)(t+5) \)

Other expressions in the original exercise like the denominator \( 9t + 36 \) was factored by extracting a common factor of 9, resulting in:

• \( 9t + 36 = 9(t+4) \)

Factoring simplifies the expressions, helping to cancel out terms and solve or manipulate algebraic equations easily.

Multiplication and Division of Fractions

Handling fractions can be made simpler by remembering a few basic rules. When you multiply fractions, you multiply the tops (numerators) and the bottoms (denominators) together:For example,

• To multiply \( \frac{(t+5)(t+4)}{9(t+4)} \) by \( \frac{9(t+5)}{t+4} \), multiply the numerators and the denominators.
• After multiplying, the expression becomes: \( \frac{(t+5)(t+4) \cdot 9(t+5)}{9(t+4) \cdot t+4} \)

For division, flip the second fraction and perform multiplication. This is called taking the reciprocal:

• Such as: \( \frac{(t+5)(t+4)}{9(t+4)} \div \frac{9(t+5)}{t+4} \) becomes \( \frac{(t+5)(t+4)}{9(t+4)} \cdot \frac{t+4}{9(t+5)} \)

This method simplifies complex rational expressions by making them easier to work with.

Simplification of Expressions

The goal of simplification is to make expressions as straightforward as possible. This involves reducing fractions by canceling common terms across the numerators and denominators. Cancellation works when the same factor appears on both sides of the fraction bar.

• Example: In the expression \( \frac{(t+5)(t+4)}{9(t+4)} \cdot \frac{9(t+5)}{t+4} \), the \( (t+4) \) and 9 can be canceled.

When we cancel these terms from both the numerator and the denominator, only the unique factors remain:

• The simplified version of part (a) becomes \( (t+5)^2 \), as derived after canceling common factors.

In part (b), the entire expression simplifies to 1 as all terms cancel each other, illustrating that simplification helps identify when expressions are equivalent to such responsive and minimal forms.