Problem 19 - Elementary Algebra

Question

For exercises 7-32, simplify. $$ \frac{p^{2}+11 p+18}{p^{2}-2 p-15} \cdot \frac{p^{2}+3 p-40}{p^{2}+10 p+16} $$

Step-by-Step Solution

Answer

\( \frac{p + 9}{p + 3} \)

Step 1: Factor the Numerators and Denominators

Factor the expressions in the numerators and denominators. Start with the first fraction: Factor the numerator: \( p^2 + 11p + 18 = (p + 9)(p + 2) \)Factor the denominator: \( p^2 - 2p - 15 = (p - 5)(p + 3) \)For the second fraction:Factor the numerator: \( p^2 + 3p - 40 = (p + 8)(p - 5) \)Factor the denominator: \( p^2 + 10p + 16 = (p + 8)(p + 2) \)

Step 2: Rewrite the Fraction Product

Using the factors obtained, rewrite the product of fractions: \[ \frac{(p + 9)(p + 2)}{(p - 5)(p + 3)} \times \frac{(p + 8)(p - 5)}{(p + 8)(p + 2)} \]

Step 3: Cancel Common Factors

Identify and cancel out common factors in the numerators and denominators: \[ \frac{(p + 9)\cancel{(p + 2)}}{(p - 5)(p + 3)} \times \frac{\cancel{(p + 8)}(p - 5)}{\cancel{(p + 8)}\cancel{(p + 2)}} \]This simplifies to: \[ \frac{(p + 9)}{(p + 3)} \times \frac{(p - 5)}{(p - 5)} \]Cancel out the common factor \((p - 5)\):\[ \frac{(p + 9)}{(p + 3)} \times 1 \]So, it reduces to: \[ \frac{(p + 9)}{(p + 3)} \]

Step 4: Simplified Expression

The simplified expression is: \( \frac{p + 9}{p + 3} \)

Key Concepts

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler polynomials that, when multiplied together, give the original polynomial. This is a crucial step in simplifying expressions because it allows us to identify and cancel out common factors.
Polynomials in the form of quadratic expressions, like those in our original exercise, usually factor into two binomials. For example:

• The quadratic expression \( p^2 + 11p + 18 \) factors into \( (p + 9)(p + 2) \).
• Similarly, \( p^2 - 2p - 15 \) is factored as \( (p - 5)(p + 3) \).

Quadratic expressions factorise by finding two numbers that multiply to give the constant term (the number without a variable) and add to give the coefficient of the middle term (the term with the power of 1). This method of factoring applies similarly to the other polynomial expressions in the exercise. Factoring helps in breaking down the problem into more manageable parts, making it easier to simplify the entire expression.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. Simplifying rational expressions involves factoring both the numerator and the denominator and then reducing the fraction by canceling out any common factors.
In our problem, we start with four quadratic polynomials (two for each fraction). After factoring each, we reassemble the product of fractions as follows: \ \( \frac{p^2+11p+18}{p^2-2p-15}\ \times\ \frac{p^2+3p-40}{p^2+10p+16} \ \). After factoring, this becomes: \ \( \frac{(p + 9)(p + 2)}{(p - 5)(p + 3)} \times \frac{(p + 8)(p - 5)}{(p + 8)(p + 2)} \ \).
Next, we'll cancel any common factors. Noticing that \((p + 2) \) and \((p - 5)\ \) appear in both the numerator and the denominator, we can cancel them out, simplifying the expression to \[ \ \frac{p + 9}{p + 3} \ \ \]. Recognizing common factors and reducing rational expressions is key to simplifying.

Simplifying Fractions

Simplifying fractions means to reduce them to their simplest form where the top and bottom of the fraction no longer have any common factors other than 1. For rational expressions, this involves canceling common polynomial factors.
In our case following the cancellation of common factors, we land with: \( \ \frac{(p + 9)}{(p + 3)}\). Here, we reach the 'simplified expression' stage where no further cancellations can be performed. This means the expression \ \ \frac{(p + 9)}{(p + 3)} \ \ is at its simplest form, and we have fully simplified the initial given expression.

• Always check for any additional factors that may simplify further.
• Ensure that both numerator and denominator are checked for common terms to cancel.

Simplifying fractions makes the equation easier to understand and use in further calculations or problem-solving, providing a cleaner result.