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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 trinomial \(6x^2 -17x + 12\) can be factorised as \( (2x-12)(3x -1)\)

Step by step solution

01

Identify the Coefficients

The given trinomial is \(6x^2 - 17x + 12\). Here the coefficient of \(x^2\) is 6, the coefficient of \(x\) is -17, and the constant term is 12.
02

Search for Factors

Find pairs of numbers that multiply to give 72 (6*12=72) and add up to -17. The pairs of numbers that multiply to 72 are (1, 72), (2, 36), (3, 24), (4, 18), (6, 12), and(8, 9). The pair (6,12) not only multiplies to give 72 but also adds up to 18. But we need the pair that adds up to -17, thus adjust the signs and pick the pair (-6,-12).
03

Rewrite the Trinomial

Rewrite the trinomial by replacing the middle term (-17x) with the terms obtained from the pair in step 2: \( 6x^2 -6x -12x + 12\).
04

Factor by Grouping

Group the first two terms and the last two terms and factor by using the greatest common factor (GCF) from each group. \(6x^2 -6x\) factors to \(2x(3x -1)\), and \(-12x + 12\) factors to \(-12(x -1)\). Combine these to get \(2x(3x -1) -12(x -1)\).
05

Factor out Common Binomial

Notice that there is a common binomial \((3x -1)\). We factor out this common binomial to obtain the final factorised form: \( (2x-12)(3x -1)\)

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