Problem 25 - Elementary Algebra

Question

For exercises 7-32, simplify. $$ \frac{m^{2}-2 m-80}{m^{2}-m-90} \cdot \frac{m^{2}+6 m-27}{m^{2}+5 m-24} $$

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1 - Factorize the Numerators and Denominators

First, factorize each polynomial in the numerators and denominators. For the first numerator: \[ m^{2} - 2m - 80 \] This can be factorized into: \[ (m - 10)(m + 8) \]For the first denominator: \[ m^{2} - m - 90 \] This can be factorized into: \[ (m - 10)(m + 9) \]For the second numerator: \[ m^{2} + 6m - 27 \] This can be factorized into: \[ (m + 9)(m - 3) \]For the second denominator: \[ m^{2} + 5m - 24 \] This can be factorized into: \[ (m + 8)(m - 3) \]

Step 2 - Rewrite the Expression

Now rewrite the expression using the factorized forms: \[ \frac{m^{2} - 2m - 80}{m^{2} - m - 90} \times \frac{m^{2} + 6m - 27}{m^{2} + 5m - 24} = \frac{(m - 10)(m + 8)}{(m - 10)(m + 9)} \times \frac{(m + 9)(m - 3)}{(m + 8)(m - 3)} \]

Step 3 - Cancel Out Common Factors

Identify and cancel out the common factors in the numerator and the denominator: Common factors are: \[ (m - 10), (m + 8), (m + 9), (m - 3) \]Therefore: \[ \frac{(m - 10)(m + 8)}{(m - 10)(m + 9)} \times \frac{(m + 9)(m - 3)}{(m + 8)(m - 3)} = \frac{\textbf{(m - 10)}(m + 8)}{\textbf{(m - 10)}(m + 9)} \times \frac{\textbf{(m + 9)}(m - 3)}{\textbf{(m + 8)}(m - 3)} = \frac{1 \times 1}{1 \times 1} = 1\]

Key Concepts

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler pieces that, when multiplied together, give the original polynomial. Think of it as reverse multiplication. For example, let's look at the polynomial \( m^2 - 2m - 80 \). By factoring it, we find its factors: \( (m - 10)(m + 8)\). This means if we expand \( (m - 10)(m + 8) \) back out, we get \( m^2 - 2m - 80 \).
To factor a polynomial, you often look for patterns or use techniques like grouping or the quadratic formula. When dealing with quadratics \( ax^2 + bx + c \), find two numbers that multiply to \( ac \) and add to \( b \).

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. Simplifying them involves factoring the polynomials and reducing the fraction. Consider our example: \( \frac{m^2-2m-80}{m^2-m-90} \cdot \frac{m^2+6m-27}{m^2+5m-24} \). After factoring the polynomials, it becomes \( \frac{(m - 10)(m + 8)}{(m - 10)(m + 9)} \cdot \frac{(m + 9)(m - 3)}{(m + 8)(m - 3)} \).
By identifying and canceling the common factors, the expression simplifies drastically. This process generally includes:

• Factoring the numerators and denominators.
• Writing the rational expression with the factors.
• Cancelling out common factors in the numerator and denominator.

Rational expressions mirror the rules of numerical fractions but involve more algebra.

Common Factors

A common factor in polynomials is a term that appears in both the numerator and the denominator. Identifying and canceling common factors is crucial because it simplifies the fraction, making it easier to work with.
For example, in the expression \( \frac{(m - 10)(m + 8)}{(m - 10)(m + 9)} \cdot \frac{(m + 9)(m - 3)}{(m + 8)(m - 3)} \), the factors \( (m - 10), (m + 8), (m + 9), \text{ and } (m - 3) \) are common between the numerator and denominator. By canceling these factors, we simplify the expression to \( 1 \).
To simplify correctly:

• Factorize all polynomials in the expression.
• Identify and cancel common factors.
• Simplify the remaining expression.

Understanding common factors helps in breaking down complex algebraic fractions into simpler forms, making them easier to handle.