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Chapter 0: Problem 33

Factor each trinomial, or state that the trinomial is prime. $$20 x^{2}+27 x-8$$

Short Answer

Expert verified
The factored form of the given trinomial \(20x^{2}+27x-8\) is \((4x-1)(5x+8)\).

Step by step solution

01

Check for common factors

First, it is important to check if all terms in the trinomial have a common factor. In this case, \(20x^2\), \(27x\), and \(-8\) do not have a common factor apart from \(1\).
02

Factor using splitting the middle term

Split the middle term in such a way that the product of two terms should be equal to \(ac\) (in the standard form of quadratic equation \(ax^2+bx+c\), \(ac = 20*(-8) = -160\) and the sum of the two terms should be equal to \(b = 27\). The two numbers that satisfy this rule are \(32\) and \(-5\). We can split the middle term \(27x\) into \(32x-5x\) to get: \(20x^2+32x-5x-8\).
03

Group and factor

Group the terms and factor by grouping: \((20x^2+32x)-(5x+8)\). This leads to \(4x(5x+8)-1(5x+8)\). Now, take the common factors from both terms: \((4x-1)(5x+8)\).

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