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Chapter 6: Problem 81

Factor completely. $$12 x^{2}+10 x y-8 y^{2}$$

Short Answer

Expert verified
The completely factored form of the trinomial \(12 x^{2}+10 x y-8 y^{2}\) would be \(2(2x-y)(3x+4y)\

Step by step solution

01

Simplify the expression

Firstly, observe that all the terms in the expression have a common factor, which is 2. So, factor it out: \(2(6x^2+5xy-4y^2)\)
02

Express the middle term as a sum of two terms

Express \(5xy\) as a sum of two terms such that the coefficients of the new terms are factors of -24, which is the product of 6 (coefficient of \(x^2\)) and -4 (coefficient of \(y^2\)). This step is required to factor by grouping. We have 8 and -3 as the two factors of -24. Therefore, rewrite \(5xy\) as \(8xy-3xy\): \(2(6x^2+8xy-3xy-4y^2)\)
03

Factor by grouping

Next step is to factor by grouping. Group the terms to have common factors: \(2[(6x^2+8xy) - (3xy+4y^2)]\). Then factor out the common factors: \(2[2x(3x+4y) - y(3x+4y)]\)
04

Factor out the common binomial factor

Finally, factor out \(3x+4y\) to get the fully factored expression: \(2(2x-y)(3x+4y)\

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Key Concepts

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

Common Factors
In algebra, a common factor is a number or expression that divides evenly into each term of a polynomial. Identifying common factors is the initial step when trying to factor a polynomial because it allows us to simplify the expression.

In the exercise provided, the polynomial is \(12x^2 + 10xy - 8y^2\). By examining each term of the expression, you can see that 2 is a common factor.

Extracting this common factor makes subsequent factoring steps easier. So, we rewrite the polynomial as:
  • \(2(6x^2 + 5xy - 4y^2)\)
This simplification is crucial because it reduces the complexity of the expression, which can aid significantly when moving on to more advanced factoring techniques.
Grouping Method
The grouping method is a powerful technique used primarily for polynomials with four or more terms. It involves rearranging and factoring the polynomial two parts at a time.

In the case of our expression, \(6x^2 + 5xy - 4y^2\), we use the split terms \(8xy\) and \(-3xy\), to rewrite the middle term \(5xy\). Therefore, the expression becomes
  • \(6x^2 + 8xy - 3xy - 4y^2\)
Next, we group the first two terms and the last two terms:
  • \((6x^2 + 8xy) + (-3xy - 4y^2)\)
This rearrangement allows us to factor each group independently. Remember to look for common factors within each group for maximum simplification.
Polynomial Factors
Finding polynomial factors is like putting a puzzle together. Each polynomial can be represented by the product of smaller expressions known as factors.

Using the expression \(2(6x^2+8xy-3xy-4y^2)\), once grouped, we factorize each part. For the first part, \(6x^2 + 8xy\), the common factor is \(2x\), which gives us:
  • \(2x(3x+4y)\)
In the second part, \(-3xy - 4y^2\), the common factor is \(-y\), resulting in:
  • \(-y(3x+4y)\)
Once the factors are collected, you can see the shared binomial factor of \(3x + 4y\), which further assists in simplifying.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations. The key to mastering them lies in understanding different factoring techniques that simplify these expressions.

In the problem \(12x^2 + 10xy - 8y^2\), through factoring, we transform complex expressions into their simpler components. Starting from identifying a common factor, then using grouping methods, we re-organize terms.

And through steps involving polynomial factors, the expression
  • \(2(2x-y)(3x+4y)\)
is uncovered as a fully factored form. Although algebraic expressions may appear daunting, recall that by breaking them down with strategies like factoring, they become much more manageable and less intimidating.

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Most popular questions from this chapter

Make Sense? In Exercises \(115-118\), determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Although I can factor the difference of squares and perfect square trinomials using trial-and-error, recognizing these special forms shortens the process.

Factor: \(9 x^{2}-16 .\) (Section 6.4, Example 1)

Use factoring to solve quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\)-intercepts. \((x-3)(x+8)=-30\)

Use factoring to solve quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\)-intercepts. \(25 x^{2}=49\)

The alligator, at one time an endangered species, is the subject of a protection program. The formula $$P=-10 x^{2}+475 x+3500$$ models the alligator population, \(P,\) after \(x\) years of the protection program, where \(0 \leq x \leq 12 .\) Use the formula to solve. After how long is the population up to \(7250 ?\)
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