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Chapter 6: Problem 1

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$x^{2}+7 x+6$$

Short Answer

Expert verified
The factorization of the trinomial \(x^{2}+7x+6\) is \((x+6)(x+1)\).

Step by step solution

01

Factorize the trinomial

Start by factoring the trinomial \(x^{2}+7 x+6\). The general form for such a trinomial is \(x^2 + bx + c\), where b is the sum of two numbers and c is their product. Here, we need to identify those two numbers that add up to 7 (the coefficient of x) and whose product is 6. The two numbers that satisfy this are 6 and 1. Thus, the trinomial \(x^{2}+7 x+6\) factors to \((x+6)(x+1)\).
02

Check the factorization using FOIL multiplication

The FOIL method is an acronym for First, Outer, Inner, and Last. This method will be used to check if the factorization was done correctly. For the trinomial above, we proceed as follows:\n\n - F (First): Multiply the first terms: \(x*x = x^{2}\).\n - O (Outer): Multiply the outer terms: \(x*1 = 1x\).\n - I (Inner): Multiply the inner terms: \(6*x = 6x\).\n - L (Last): Multiply the last terms: \(6*1 = 6\).\n\n If you add these all together, you get \(x^{2}+7x+6\), which is the initial trinomial, meaning the factorization was correct.
03

Final Answer

After factoring and confirming using the FOIL method, we can confirm that the factorization of the trinomial \(x^{2}+7 x+6\) is \((x+6)(x+1)\).

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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. $$20 y^{2}+12 y+1$$

Factor completely. $$10 x^{2}(x+1)-7 x(x+1)-6(x+1)$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$16 a^{2}-32 a b+12 b^{2}$$

Describe some of the difficulties in factoring polynomials. What suggestions can you offer to overcome these difficulties?

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$3 x^{4} y^{2}-3 x^{2} y^{2}$$
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