Problem 20 - Intermediate Algebra

Question

Multiply, and then simplify, if possible. See Example 2. $$ \frac{3 p^{2}}{6 p+24} \cdot \frac{p^{2}-16}{6 p} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{p(p-4)}{12} \).

Step 1: Factorize the Numerators and Denominators

First, we observe that the numerators and denominators might have common factors that can be canceled out. The expression is \( \frac{3p^{2}}{6p+24} \cdot \frac{p^{2}-16}{6p} \). Start by factorizing:Numerator 1: \( 3p^2 \) doesn't need factoring.Denominator 1: \( 6p + 24 \) can be factorized to \( 6(p + 4) \).Numerator 2: \( p^2 - 16 \) is a difference of squares, which can be factorized to \( (p - 4)(p + 4) \).Denominator 2: \( 6p \) is already simplified.

Step 2: Write the Factored Expression

Substitute the factorized terms back into the original expression:\[\frac{3p^{2}}{6(p+4)} \cdot \frac{(p - 4)(p + 4)}{6p}\]

Step 3: Cancel Out Common Factors

Now, look for common factors between numerators and denominators that can be canceled:- Both numerators and denominators contain a factor of 3 in the terms \( 3p^{2} \) and \( 6 \).- \( p+4 \) appears in both a numerator and a denominator, so they can be canceled.- After canceling, the simplified expression looks like this:\[\frac{p^{2}(p-4)}{12p}\]- Further simplify by canceling \( p \) in \( p^{2} \) and \( 12p \):\[\frac{p(p-4)}{12}\]

Step 4: Write Final Simplified Expression

Thus, the final simplified expression is:\[\frac{p(p-4)}{12}\]

Key Concepts

Factoring Polynomials

Factoring polynomials is like finding the pieces that multiply together to give the original polynomial. This technique is especially useful for simplifying and solving equations. In this exercise, recognizing different forms of polynomials helped us in factorization.

For instance, the expression \( p^2 - 16 \) is a special type known as the difference of squares. Here, it can be expressed as two binomials: \((p - 4)(p + 4)\).

• When you see a polynomial squared minus a constant squared, think about this kind of factorization.
• It follows the generic pattern: \( a^2 - b^2 = (a - b)(a + b) \).

Another example is \( 6p + 24 \), which can be simplified by extracting a common factor:

• Since both terms are divisible by 6, it factors to \( 6(p + 4) \).
• Look for the greatest common factor (GCF) and divide each term by it.

Simplifying Rational Expressions

Simplification of rational expressions involves reducing the expression to its simplest form. This can be accomplished by canceling out common factors from the numerator and denominator.

In our exercise, once we factorized the expression, we looked for and canceled common elements:

• The expression turned into \( \frac{3p^{2}}{6(p+4)} \cdot \frac{(p - 4)(p + 4)}{6p} \).
• \( p + 4 \) was present in both a numerator and a denominator, which allowed it to be canceled.
• Both numerator terms contained a factor of 3, simplifying the expression further.

Further simplification occasionally involves like term cancellation, which is what happened with \( p^2 \) and \( 12p \), resulting in \( \frac{p(p-4)}{12} \). Here, dividing out a common \( p \) reduced the complexity of the expression significantly.

Multiplying Rational Expressions

To multiply rational expressions, multiply the numerators together and the denominators together, as you would with fractions. To simplify the resulting expression, factor and reduce if possible.

In our task, we multiplied:

• Numerators: \( 3p^{2} \cdot (p - 4)(p + 4) \).
• Denominators: \( 6(p + 4) \cdot 6p \).

After multiplication, the process was about simplification:

• Factored terms make it easier to see which parts can be canceled. Factorized forms revealed that common terms could be removed before final multiplication.
• Canceling common terms reduced it down to \( \frac{p(p-4)}{12} \).

Canceling before multiplying reduces chances of error and makes ultimate simplification easier. This methodical approach brings an otherwise complex expression down to a neat, easy-to-use form.