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Chapter 0: Problem 73

Factor completely, or state that the polynomial is prime. $$2 x^{2}-2 x-112$$

Short Answer

Expert verified
The factored form of the polynomial \(2 x^{2}-2 x-112\) is \(2(x - 8)(x + 7)\).

Step by step solution

01

Identify the coefficients and constant

The given polynomial is \(2 x^{2}-2 x-112\). Here, the coefficient of \(x^2\) is 2, the coefficient of x is -2, and the constant term is -112.
02

Factor out the common factor

We can see that each term in the polynomial can be divided by 2 which is a common factor to all terms. Taking 2 common out, we get \(2(x^{2}-x-56)\). Now, we have to focus on factoring the expression within the parenthesis, \(x^{2}-x-56\).
03

Find the factors of 56 that sum to -1

We look for two numbers that multiply to be -56 (the product of a and c in the quadratic equation) and add to -1 (the coefficient of the x-term). The numbers -8 and 7 fit this requirement since -8*7 = -56 and -8+7 = -1.
04

Rewrite the polynomial

We can now rewrite our polynomial by replacing -x with -8x + 7x. This gives us \(2(x^{2}- 8x + 7x - 56)\). The expression within the parenthesis now becomes \(2((x^{2}- 8x) + (7x - 56))\). We can factor an x from the first two terms and a 7 from the second two terms, yielding \(2(x(x - 8) + 7(x - 8))\).
05

Extract the common binomial

We can now see that we have a common binomial factor of (x - 8). Taking (x - 8) common out we get \(2(x - 8)(x + 7)\)

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Most popular questions from this chapter

Will help you prepare for the material covered in the first section of the next chapter. If \(y=4-x^{2},\) find the value of \(y\) that corresponds to values of \(x\) for each integer starting with -3 and ending with 3

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Determine whether each statement makes sense or does not make sense, and explain your reasoning. I subtracted \(\frac{3 x-5}{x-1}\) from \(\frac{x-3}{x-1}\) and obtained a constant.
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