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Chapter 5: Problem 71

Factor 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

01

Identify the Common Factor

Observe the polynomial ((2 p+q) r^{2}-12(2 p+q) r+27(2 p+q)} and identify ((2 p+q)} as the common factor in all the terms.
02

Factor out the Common Term

Factor out the common term ((2 p+q)} from the polynomial: (2 p+q) [r^{2}-12r+27]})
03

Factor the Quadratic Expression

Factor the quadratic expression (r^{2}-12r+27)}. Look for two numbers whose product is (27)} and whose sum is (-12)}: (-3)} and (-9)}. Thus, we can write (r^{2}-12r+27)} as (r-3)(r-9)}.
04

Final Factored Form

Combine the factored terms to get the final factored form of the polynomial: ((2 p+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.

Common Factor
When we are given a polynomial to factor, the first step is always to identify if there is a common factor in all terms. A common factor is any term that is present in each part of the polynomial. For example, in the polynomial \((2p + q) r^{2} - 12(2 p + q) r + 27(2 p + q)\), the common factor is \((2 p + q)\).
This means that every term in this polynomial has \((2 p + q)\).
Finding the common factor simplifies the polynomial and makes it easier to work with.
By factoring out \((2 p + q)\), we reduce our problem significantly.
Quadratic Expression
After factoring out the common factor, we are often left with a simpler expression. In this case, after removing \((2 p + q)\) from \((2 p + q) r^{2} - 12(2 p + q) r + 27(2 p + q)\), we get \((r^{2} - 12r + 27)\).
This is a quadratic expression, which is an equation of the form \[ax^{2} + bx + c\].
Quadratics are easily recognizable by their \(x^{2}\) term. Learning to work with quadratic expressions is critical because they show up frequently in algebra.
Factored Form
Factoring a polynomial means writing it as a product of simpler expressions. Once we have the quadratic expression \((r^{2} - 12r + 27)\), our goal is to write it in a factored form like \((r-a)(r-b)\).
Getting to the factored form involves finding numbers \(a\) and \(b\) that satisfy certain conditions tied to the original coefficients of the quadratic.
This process helps simplify solving polynomial equations and analyzing their properties.
Product-Sum Method
To factor the quadratic expression further, we use the product-sum method. This involves finding two numbers that multiply to the constant term (the product) and add up to the coefficient of the \(r\) term (the sum).
For \((r^{2} - 12r + 27)\), we look for numbers that multiply to 27 and add up to -12. These numbers are -3 and -9. This allows us to rewrite the quadratic as \((r - 3)(r - 9)\).
Using this method makes factoring straightforward and feasible, turning a challenging polynomial into a solvable expression.

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Most popular questions from this chapter

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