Problem 44 - Elementary Algebra

Question

For exercises \(5-48\), simplify. $$ \frac{8 p^{2}}{4 p^{2}+32 p+64}-\frac{128}{4 p^{2}+32 p+64} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{2(p - 4)}{p + 4} \).

Step 1 - Identify the common denominator

The common denominator in the expression is the quadratic polynomial \(4p^2 + 32p + 64\).

Step 2 - Factor the denominator

Factor \(4p^2 + 32p + 64\) as follows: \(4p^2 + 32p + 64 = 4(p^2 + 8p + 16)\). Recognizing this as a perfect square trinomial, it becomes: \( = 4(p + 4)^2 \).

Step 3 - Simplify the first fraction

Rewrite the first fraction with the factored denominator: \( \frac{8p^2}{4(p + 4)^2} \). Cancel common factors. The 4 in the denominator divides into the 8 in the numerator, leaving: \( \frac{2p^2}{(p + 4)^2} \).

Step 4 - Simplify the second fraction

Rewrite the second fraction with the factored denominator: \( \frac{128}{4(p + 4)^2} \). Cancel common factors. The 4 in the denominator divides into the 128 in the numerator, leaving: \( \frac{32}{(p + 4)^2} \).

Step 5 - Combine the fractions

Combine the simplified fractions: \( \frac{2p^2}{(p + 4)^2} - \frac{32}{(p + 4)^2} = \frac{2p^2 - 32}{(p + 4)^2} \).

Step 6 - Factor the numerator if possible

Factor the numerator \(2p^2 - 32\): \( 2(p^2 - 16) = 2(p + 4)(p - 4) \).

Step 7 - Simplify the final expression

Substitute back into the fraction to get: \( \frac{2(p + 4)(p - 4)}{(p + 4)^2} \). Cancel the common \((p + 4)\) factors, leaving: \( \frac{2(p - 4)}{p + 4} \).

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra. It involves breaking down a polynomial into simpler elements called factors that, when multiplied together, give the original polynomial.
To factor a polynomial like the denominator in our problem, you need to find common factors and apply factoring techniques.
In the given problem, the quadratic polynomial is factored by finding the greatest common factor (GCF) first. We factored out 4 from all terms in the polynomial: \[ 4p^2 + 32p + 64 = 4(p^2 + 8p + 16) \].
Then, we recognized that the remaining polynomial inside the parentheses is a perfect square trinomial. We continue to break it down to \[ 4(p + 4)^2 \].
Using these techniques correctly makes solving algebraic fractions easier.

Common Denominators

Finding a common denominator is essential when dealing with fraction addition and subtraction. A common denominator is a shared multiple of the denominators of the fractions involved.
In our case, both fractions \( \frac{8p^2}{4(p+4)^2} \text{ and } \frac{128}{4(p+4)^2} \) already have a common denominator.
Having a common denominator allows us to easily subtract the fractions by combining the numerators while keeping the same denominator. This simplification process streamlines the expression and makes it easier to solve.

Simplifying Expressions

Simplifying expressions involves reducing them to their most compact and understandable form. After factoring the denominator and recognizing the common denominator, we simplify by canceling common factors.
First, we rewrite the fractions: \( \frac{8p^2}{4(p + 4)^2} \rightarrow \frac{2p^2}{(p + 4)^2} \) and \( \frac{128}{4(p + 4)^2} \rightarrow \frac{32}{(p + 4)^2} \).
Next, combining them results in \( \frac{2p^2}{(p + 4)^2} - \frac{32}{(p + 4)^2} \rightarrow \frac{2p^2 - 32}{(p + 4)^2} \).
Lastly, we factor and simplify the numerator for the final simplified form.

Perfect Square Trinomials

Perfect square trinomials take a specific and recognizable form. They are of the type \( a^2 + 2ab + b^2 \) and can be factorized to \( (a+b)^2 \).
In our problem, \( 4p^2 + 32p + 64 \rightarrow 4(p^2 + 8p + 16) \).
The polynomial inside the parentheses \( p^2 + 8p + 16 \) is recognized as a perfect square trinomial. Factoring it, we get \( 4(p+4)^2 \).
Recognizing and factoring perfect square trinomials can be a powerful tool in simplifying complex algebraic fractions.

Fraction Subtraction

Fraction subtraction is simplified by combining fractions with a common denominator, then subtracting their numerators.
In our problem, we have: \( \frac{2p^2}{(p + 4)^2} - \frac{32}{(p + 4)^2} \).
Since the denominators are the same, subtracting the fractions simplifies to: \( \frac{2p^2 - 32}{(p + 4)^2} \).
Further factoring and simplification lead to: \( \frac{2(p + 4)(p - 4)}{(p + 4)^2} \).
Canceling the common terms, the final simplified form is \( \frac{2(p - 4)}{p + 4} \).
Understanding these subtraction techniques and simplifications makes fraction operations more manageable.