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Chapter 6: Problem 83

Factor completely. $$8 x^{2} y+34 x y-84 y$$

Short Answer

Expert verified
The factored form of the expression is \(y(2x + 7)(4x + 6)\).

Step by step solution

01

Identifying Common Factor

Start by identifying the common factors. In this expression, the terms \(8x^{2}y\), \(34xy\), and \(-84y\) all have \(y\) as a common factor.
02

Factoring Out the Common Factor

After identifying the common factor, the next step is to factor it out. Factoring out \(y\) from \(8x^{2}y+34xy-84y\) gives \(y(8x^{2}+34x-84)\).
03

Factoring the Quadratic

Now, factor the quadratic \(8x^{2} + 34x - 84\) by looking for two numbers that multiply to \(-672 = 8*(-84)\) and add up to \(34\). Those numbers are \(28\) and \(24\), so the breaking down the middle term gives \((8x^{2} + 28x + 6x - 84)\) which can be factored by grouping. Grouping gives \(y(4x(2x + 7) + 6(2x + 7))\).
04

Factor by Grouping

After factoring the quadratic by grouping, we obtain \(y(2x + 7)(4x + 6)\). This is the completely factored form of given polynomial.

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Key Concepts

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

Common Factor
A common factor in polynomials is a shared element among all terms in an expression. Finding this simplifies the polynomial and makes further factoring easier. In the expression \(8x^2y + 34xy - 84y\), you can see that \(y\) is present in each term. Recognizing this helps you factor it out initially.
Factoring out the common terms:
  • Look at all the terms in the polynomial and identify the highest common factor in terms of variables and coefficients.
  • Here, \(y\) was the obvious common factor across all terms.
  • By factoring out \(y\), the expression gets simplified to \(y(8x^2 + 34x - 84)\).
This step reduces the complexity of the polynomial and prepares it for more advanced factoring methods, such as quadratic factoring.
Quadratic Factoring
Quadratic factoring involves breaking down a quadratic expression of the form \(ax^2 + bx + c\) into two binomials.
This comes in handy while dealing with the simplified expression from the previous step: \(8x^2 + 34x - 84\). The goal is to express it in the form \((px + q)(rx + s)\).
  • Calculate the product \(a \cdot c\), which is \(8 \times -84 = -672\).
  • Look for two numbers that multiply to \(-672\) and add to \(34\). In this case, those numbers are \(28\) and \(-24\).
  • Rewrite the expression by breaking the middle term: \(8x^2 + 28x + 6x - 84\).
This punts the expression into a form that's approachable by using the next method: factoring by grouping.
Factoring by Grouping
Factoring by grouping is a method used to simplify complex polynomials by creating pairs of terms that can be factored easily. It proves effective particularly after using quadratic factoring on complicated terms.
  • First, take the expression \(8x^2 + 28x + 6x - 84\) from quadratic factoring.
  • Group the terms to form two binomial pairs: \((8x^2 + 28x)\) and \((6x - 84)\).
  • Factor each binomial: from the first, take out \(4x\) to get \(4x(2x + 7)\); from the second, take out \(6\) to get \(6(2x + 7)\).
Now the expression looks like \(y(4x(2x + 7) + 6(2x + 7))\). Both parts feature \(2x + 7\) as a common factor, allowing for further factoring as \(y(2x + 7)(4x + 6)\).
This systematic breakdown makes even a complex polynomial much more palatable and manageable to work with.

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Most popular questions from this chapter

A vacant rectangular lot is being turned into a community vegetable garden measuring 15 meters by 12 meters. A path of uniform width is to surround the garden. If the area of the lot is 378 square meters, find the width of the path surrounding the garden.

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Use the \(x\)-intercepts for the graph in \(a[-10,10,1]\) by \([-13,10,1]\) viewing rectangle to solve the quadratic equation. Check by substitution. Use the graph of \(y=x^{2}+3 x-4\) to solve \(x^{2}+3 x-4=0\).

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