Problem 81 Factor each polynomial. $$ (... [FREE SOLUTION]

Chapter 5: Problem 81

Factor each polynomial. $$ (2 p+q) r^{2}-12(2 p+q) r+27(2 p+q) $$

Short Answer

Expert verified

(2p + q)(r - 3)(r - 9)

Step by step solution

Identify the Greatest Common Factor (GCF)

First, identify the common factor in each term of the polynomial. In this case, notice that each term contains $ (2p + q) $ as a factor.

Factor Out the GCF

Factor out $ (2p + q) $ from each term of the polynomial: \[ (2p + q)(r^2 - 12r + 27) \]

Factor the Quadratic Expression

Next, factor the quadratic expression inside the parentheses. Find two numbers that multiply to 27 and add to -12. These numbers are -3 and -9. Therefore, \[ r^2 - 12r + 27 = (r - 3)(r - 9) \]

Combine All Factors

Combine the factored terms: \[ (2p + q)(r - 3)(r - 9) \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the highest number that divides exactly into two or more numbers. In algebra, it's the largest expression that divides all terms in a polynomial.
To identify the GCF:

List out the factors of each term.
Find the largest common factor from these lists.

For example, in the polynomial equation $(2p+q)r^2 -12(2p+q)r + 27(2p+q)$, we can see that each term contains $(2p+q)$ as a factor. Therefore, the GCF is $(2p+q)$. Factoring out the GCF simplifies the task of further solving or factoring the polynomial.

Factoring Quadratic Expressions

Factoring quadratic expressions is a crucial step in polynomial factoring. A quadratic expression takes the form $ax^2 + bx + c$. To factor it:

First, identify the product $ac$ and the middle coefficient $b$.
Find two numbers that multiply to $ac$ and add to $b$.
Rewrite the quadratic expression using these two numbers and factor by grouping.

For example, consider $r^2 - 12r + 27$. Here, $a = 1$, $b = -12$, and $c = 27$. We need two numbers that multiply to 27 and add to -12, which are -3 and -9. Hence, we can write: $r^2 - 12r + 27 = (r - 3)(r - 9)$.

Polynomial Factoring Steps

Factoring a polynomial often involves several clear steps:

Identify the Greatest Common Factor (GCF) in all terms of the polynomial.
Factor out the GCF from each term, simplifying the polynomial.
Factor any remaining quadratic expressions by finding two numbers that multiply to the constant term and add to the middle coefficient.
Combine all factors to get the factored form of the polynomial.

As seen in the example $(2p+q)r^2 - 12(2p+q)r + 27(2p+q)$, these steps yield:
1. Factor out $(2p + q)$: $(2p + q)(r^2 - 12r + 27)$.
2. Factor the quadratic: $r^2 - 12r + 27 = (r - 3)(r - 9)$.
3. Combine all factors: $(2p + q)(r - 3)(r - 9)$.

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91影视

Factor each polynomial. $$ (2 p+q) r^{2}-12(2 p+q) r+27(2 p+q) $$

Short Answer

Step by step solution

Identify the Greatest Common Factor (GCF)

Factor Out the GCF

Factor the Quadratic Expression

Combine All Factors

Key Concepts

Greatest Common Factor (GCF)

Factoring Quadratic Expressions

Polynomial Factoring Steps

One App. One Place for Learning.

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