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Chapter 0: Problem 32

Factor each trinomial, or state that the trinomial is prime. $$9 x^{2}+5 x-4$$

Short Answer

Expert verified
The factored form of the given trinomial is \((9x - 4)(x + 1)\).

Step by step solution

01

Identify the Coefficients

The coefficients of the trinomial are \(a = 9\), \(b = 5\) and \(c = -4\).
02

Multiplication and Addition Check

Look for two integers that multiply to give -36 (i.e., \(9*-4\)) and add to give 5 (the middle coefficient). After testing some pairs, you'll find that 9 and -4 work because their product is -36 and their sum is 5.
03

Factor the Trinomial

Now replace the middle term of the trinomial with two terms. The original equation, \(9x^2 + 5x - 4\), becomes \(9x^2 + 9x - 4x - 4\). Now, group the terms into two binomials. This gives us \(9x^2 + 9x\) and \(-4x - 4\). Factor by GCF (Greatest Common Factor) from each binomial. The equation becomes \(9x(x + 1) - 4(x + 1)\). Notice how \(x + 1\) is a common factor. This gives the factored form of the trinomial as \((9x - 4)(x + 1)\).

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