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Chapter 6: Problem 4

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$x^{2}+9 x+14$$

Short Answer

Expert verified
The factors of the trinomial \(x^2 + 9x + 14\) are \((x + 2)\) and \((x + 7)\).

Step by step solution

01

Identifying coefficients

The first step is to recognize the coefficients in the quadratic trinomial \(x^2 + 9x + 14\). In this case, \(a = 1\), \(b = 9\), and \(c = 14\).
02

Finding the factors of 'c'

The next step is to find two numbers that multiply to give \(c = 14\) and add up to give \(b = 9\). These numbers are \(2\) and \(7\) since \(2*7 = 14\) and \(2+7 = 9\).
03

Writing the factorization

The factors of the trinomial are \(x + 2\) and \(x + 7\). Therefore, we can rewrite the trinomial as \((x + 2)(x + 7)\).
04

Verifying the solution

Use the FOIL method to multiply the factors back together to make sure they form original trinomial. \(x* x = x^2\), \(x*7 = 7x\), \(2*x = 2x\), and \(2*7 = 14\). If we add these together, we get \(x^2 + 7x + 2x + 14\) which simplifies to \(x^2 + 9x + 14\), which is our original trinomial, so the factorization is correct.

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