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Chapter 0: Problem 28

Factor each trinomial, or state that the trinomial is prime. $$6 x^{2}-17 x+12$$

Short Answer

Expert verified
The factored form of the trinomial \(6x^2 - 17x + 12\) is \((2x - 3)(3x - 4)\)

Step by step solution

01

Identify the coefficients and constant

The coefficients of the trinomial \(6x^2 - 17x + 12\) are \( a = 6 \), \( b = -17 \), and \( c = 12 \).
02

Find two numbers that add up to -17 and multiply to 72

We must find two numbers, p and q, such that \( p+q = b = -17 \) and \( p*q = a*c = (6)(12) = 72 \). After trying a few possibilities, we find that -9 and -8 fit these conditions because -9 + -8 = -17 and -9*(-8) = 72.
03

Break the middle term

Split the original trinomial by breaking the middle term into two terms using the two numbers we found: \(6x^2 - 9x - 8x + 12\)
04

Factor by grouping

Apply the factor by grouping method: Group the first two terms together and the last two terms together, and factor out the greatest common factor (GCF) from each binomial. \[6x^2 - 9x - 8x + 12 = 3x(2x - 3) - 4(2x - 3)\] Notice that \(2x - 3\) is a common factor.
05

Combine like terms

Take out the common binomial term to obtain the factored form of the trinomial: \[(2x - 3)(3x - 4)\]

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Most popular questions from this chapter

will help you prepare for the material covered in the first section of the next chapter. If \(y=4-x^{2},\) find the value of \(y\) that corresponds to values of \(x\) for each integer starting with \(-3\) and ending with 3 .

Factor completely. $$x^{4}-y^{4}-2 x^{3} y+2 x y^{3}$$

Simplify by reducing the index of the radical. $$ \sqrt[9]{x^{6} y^{3}} $$

Factor Completely. $$6 x^{4}+35 x^{2}-6$$

Determine whether each statement is trueor false. If the statement is false, make the necessary change(s) toproduce a true statement. \(x^{4}-16\) is factored completely as \(\left(x^{2}+4\right)\left(x^{2}-4\right)\)
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