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Chapter 0: Problem 27

Factor each trinomial, or state that the trinomial is prime. $$6 x^{2}-11 x+4$$

Short Answer

Expert verified
The factored form of the trinomial \(6x^{2} - 11x + 4\) is \((3x - 4)(2x - 1)\).

Step by step solution

01

Analyzing the Trinomial

In the trinomial \(6x^{2} - 11x + 4\), observe that the coefficient of \(x^{2}\) is 6, the coefficient of \(x\) is -11, and the constant is 4.
02

Multiplying the Coefficients of the First and Third Terms

Multiply the coefficients of the first term and the third term. The result is \(6*4 = 24\).
03

Finding Two Numbers

Find two numbers that multiply together to give 24, and add up to -11. The numbers -8 and -3 meet these conditions because \(-8*-3 = 24\) and \(-8 + -3 = -11\).
04

Rewriting the Middle Term

Split the middle term, -11x, into two terms using the numbers found in the previous step. This gives us \(6x^{2} -8x -3x + 4\).
05

Factoring by Grouping

Factor by grouping. This involves separating the trinomial into two binomials, and factoring out the greatest common factor (GCF) from each binomial. First binomial is \(2x(3x - 4)\) and second binomial is \(-1(3x - 4)\).
06

Final Factoring

Observe that the terms in brackets are the same. Factor out the common binomial. The result is \((3x - 4)(2x - 1)\).

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