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Chapter 0: Problem 24

Factor each trinomial, or state that the trinomial is prime. $$2 x^{2}+5 x-3$$

Short Answer

Expert verified
So, the factored form of \(2x^2 + 5x - 3\) is \((2x + 3)(x - 1)\).

Step by step solution

01

Identify the Trinomial

The given trinomial is \(2x^2 + 5x - 3\) where \(a = 2\), \(b = 5\), and \(c = -3\). We will factor this trinomial.
02

Determine the Products

Find two numbers that multiply to \(ac\) (which is \(2 * -3 = -6\)) and add up to \(b\) (which is 5). These two numbers are 3 and -2.
03

Rewrite middle term

Rewrite the middle term (\(5x\)) of the trinomial as the sum of the two numbers multiplied by \(x\), \(3x\) and \(-2x\). So, \(2x^2 + 5x - 3\) becomes \(2x^2 + 3x - 2x - 3\).
04

Factor-by-grouping

Apply 'factor-by-grouping' to the expression. To do this, group the first two terms and the last two terms together: \((2x^2 + 3x) - (2x + 3)\) . Then, factor out the greatest common factor from each group: \(x(2x + 3) - 1(2x + 3)\).
05

Complete factoring

Notice that \((2x + 3)\) is a common binomial term of both terms. Factor out the common binomial, which gives the final factored form of the trinomial: \((2x + 3)(x - 1)\).

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