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Chapter 6: Problem 5

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$x^{2}+11 x+10$$

Short Answer

Expert verified
The factorization of the trinomial \(x^{2}+11 x+10\) is \((x+10)(x+1)\).

Step by step solution

01

- Factorize the Trinomial

The factorization process involves identifying two numbers that both add up to the coefficient of the 'x' term (which is +11 in this case), and multiply to give the constant term (which is +10). These two numbers are 10 and 1, because 10+1 = 11 and 10*1 = 10. Therefore, the factorization of \(x^{2} + 11x + 10\) is \( (x+10)(x+1)\).
02

- Check Factorization Using FOIL

To verify the factorization, apply the FOIL method (First, Outside, Inside, Last) to the factored form. For the two binomials \((x+10)\) and \((x+1)\). First terms: x*x = \(x^2\); Outside terms: 10*x = 10x; Inside terms: 1*x = x; Last terms: 10*1 = 10. Add all these together to get \(x^{2} + 11x + 10\), which matches the original trinomial and confirms that the factorization is correct.

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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. $$2 r^{3}+30 r^{2}-68 r$$

Use the \([\mathrm{GRAPH}]\) or \([\mathrm { TABLE }]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$\begin{aligned} &x^{4}-16=\left(x^{2}+4\right)(x+2)(x-2) ;[-5,5,1] \text { by }\\\ &[-20,20,2] \end{aligned}$$

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. $$-7 y^{3}+21 y^{2}-14 y$$

The polynomial \(4 x^{2}+100\) is the sum of two squares and therefore cannot be factored. If the general factoring strategy is used to factor a polynomial, at least two factorizations are necessary before the given polynomial is factored completely.

Factor completely. $$7 x^{4}+34 x^{2}-5$$
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