Problem 67 - Elementary Algebra

Question

For exercises 39-82, simplify. $$ \frac{p^{2}-64}{p+4} \div \frac{p+8}{-p-4} $$

Step-by-Step Solution

Answer

\(\frac{-(p - 8)}{p + 4}\) or \(\frac{8 - p}{p + 4}\)

Step 1: Factor the Numerator

First, factor the numerator of the first fraction. The numerator is given by \(p^2 - 64\). Notice that this is a difference of squares. Using the formula \(a^2 - b^2 = (a - b)(a + b)\), we can factor it as:\[p^2 - 64 = (p - 8)(p + 8)\]

Step 2: Rewrite the Expression

Rewrite the original expression using the factored form:\[\frac{(p - 8)(p + 8)}{p + 4} \times \frac{-p - 4}{p + 8}\]

Step 3: Simplify the Expression

Now, simplify the expression by canceling common factors in the numerator and the denominator. Notice that \(p + 8\) appears in both the numerator of the first fraction and the denominator of the second fraction. Also, \(p + 4\) and \(-p - 4\) are opposites.So, the expression simplifies as:\[\frac{(p - 8)(p + 8)}{p + 4} \times \frac{-p - 4}{p + 8} = \frac{(p - 8)}{p + 4} \times (-1)\]

Step 4: Multiply the Simplified Expression

Now, multiply the remaining expressions:\[\frac{(p - 8)}{p + 4} \times (-1) = \frac{-(p - 8)}{p + 4} = \frac{-p + 8}{p + 4}\]

Key Concepts

Difference of squares

When you see a polynomial like \(p^2 - 64\), it might look complicated, but it's actually a special form called a difference of squares. A difference of squares is when you have two perfect squares separated by a subtraction sign. The general form is \(a^2 - b^2\). This can be factored into \((a - b)(a + b)\). So, for \(p^2 - 64\), we can write it as \(p^2 - 8^2\). Then, by using the formula, it becomes \((p - 8)(p + 8)\). Recognizing this pattern helps simplify many algebraic expressions.

Factoring polynomials

Factoring polynomials is an essential skill in algebra. This process involves breaking down a polynomial into simpler 'factor' polynomials that, when multiplied together, give you the original polynomial. For instance, with \(p^2 - 64\), we see it's a difference of squares and factor it into \((p - 8)(p + 8)\). When you factor polynomials properly, it opens up opportunities to simplify and solve more complex expressions further down the line. Always look for common patterns like the difference of squares, common factors, or trinomials.

Canceling common factors

Once a polynomial is factored, you can simplify expressions by canceling common factors. Take the expression \(\frac{(p - 8)(p + 8)}{p + 4} \times \frac{-p - 4}{p + 8}\). Notice that \(p + 8\) appears in both the numerator of the first fraction and the denominator of the second. We can cancel these out. Also, observe that \(p + 4\) and \(-p - 4\) are opposites, meaning they cancel out each other leaving a factor of -1. This step sharply reduces the complexity of the expression.

Multiplying fractions

Multiplying fractions involves multiplying the numerators together and the denominators together. After canceling common factors, we have: \(\frac{(p - 8)}{p + 4} \times (-1)\). We can multiply the numerator by -1 directly, resulting in \(\frac{-(p - 8)}{p + 4}\). Whenever you multiply fractions, simplify first by canceling out any common terms in the numerators and denominators before multiplying, to make the calculations easier.

Algebraic simplification

Algebraic simplification is the process of reducing an expression to its simplest form. Let’s revisit the expression \(\frac{-(p - 8)}{p + 4}\). This can be further simplified to \(\frac{-p + 8}{p + 4}\). Simplifying algebraic expressions often involves combining like terms, factoring, and canceling. By breaking down complex expressions step-by-step and using algebraic rules and properties, we can make these expressions more manageable and easier to understand. Simplifying not only makes the equation look cleaner but also helps in solving problems more efficiently.