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Chapter 6: Problem 14

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$x^{2}+3 x-28$$

Short Answer

Expert verified
The factored form of the trinomial \(x^{2}+3x-28\) is \((x+7)(x-4)\).

Step by step solution

01

Define the Trinomial

The trinomial we are working on is \(x^{2}+3x-28\).
02

Factoring the Trinomial

We need to find two numbers that multiply to -28 (the constant term), and add to 3 (the coefficient of the x term). After consideration, 7 and -4 fit these criteria. Thus, the factored form of the trinomial is \((x+7)(x-4)\).
03

Checking the Factorization Using FOIL

Let us use the FOIL method to expand the factored form and confirm that we obtain the original trinomial: \nFirst terms: \(x*x = x^{2}\), \nOuter terms: \(x*(-4) = -4x\), \nInner terms: \(7*x= 7x\), \nLast terms: \(7*(-4) = -28\).\nAdding these together we obtain \(x^{2}+3x-28\), which matches the given trinomial.

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Most popular questions from this chapter

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$y^{3}+2 y^{2}-y-2$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(x^{2}-9=(x-3)^{2}\) for any real number \(x\)

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$2 b x^{2}+44 b x+242 b$$

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$9 x^{4}+18 x^{3}+6 x^{2}$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$81 x^{4} y-y^{5}$$
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