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Chapter 6: Problem 9

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$x^{2}-12 x+36$$

Short Answer

Expert verified
The trinomial \(x^2 - 12x + 36\) can be factorized as \((x - 6)^2\).

Step by step solution

01

Identifying coefficients

First, identify the coefficients of the trinomial which is written in the form \(ax^{2}+bx+c\). We find that \(a = 1\), \(b = -12\), and \(c = 36\).
02

Check for Perfect Square trinomial and Factorizing

The square of -6 is 36 and twice of 6 is 12. This means the trinomial is a perfect square. Following the expansion \((a-b)^2 = a^2 - 2ab + b^2\), we find that \(a = x\) and \(b = 6\). So, the factorized form of the trinomial \(x^2 -12x + 36\) is \((x-6)^2\).
03

Verify with FOIL method

Verification of the factorized form is done using FOIL (First, Outer, Inner, Last) method. Multiply \((x-6) * (x-6)\): First terms: \(x * x = x^2\); Outer terms: \(x * -6 = -6x\); Inner terms: \(-6 * x = -6x\); Last terms: \(-6 * -6 = 36\). Adding these, we recover the original trinomial \(x^2 -12x + 36\). This verifies our factorization.

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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. $$3 x^{2}+243$$

Factor completely. $$x^{2}+8 x+16-25 a^{2}$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$x^{2}-4 x y-12 y^{2}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The polynomial \(4 x^{2}+100\) is the sum of two squares and therefore cannot be factored.

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$25 a^{2}+25 a b+6 b^{2}$$
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