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Chapter 6: Problem 8

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$x^{2}-13 x+40$$

Short Answer

Expert verified
The factorization of the trinomial \(x^{2} - 13x + 40\) is \((x-8)(x-5)\). The correctness of this factorization is confirmed using the FOIL method.

Step by step solution

01

Factor the trinomial

The first action is to find two numbers which multiply to give 40 (the last term of the trinomial) and also add up to -13 (the coefficient of the middle term). Those numbers are -8 and -5 because -8*-5 = 40 and -8 + -5 = -13. Now, the trinomial can be factored as:\[x^{2}-13 x+40 = (x-8)(x-5)\]
02

Verify the factoring with the FOIL method

Verifying will mean doing the inverse of factoring, which is expanding. This is done by applying the FOIL method. We multiply the First terms, the Outer, the Inner, and finally the Last terms: First terms: \(x \cdot x = x^{2}\)Outer terms: \(x \cdot -5 = -5x\)Inner terms: \(-8 \cdot x = -8x\)Last terms: \(-8 \cdot -5 = 40\)Adding these up results in \(x^{2} - 13x + 40\), which is the original trinomial, verifying that our factorization is correct.

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