In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. 9x²+5x−4

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Identify the trinomial: $9x^2 + 5x - 4$. The goal is to factor it into the form $(ax + b)(cx + d)$, where $a$, $b$, $c$, and $d$ are constants.

Multiply the leading coefficient ($9$) by the constant term ($-4$). This gives $9 \times -4 = -36$. Now, find two numbers that multiply to $-36$ and add to the middle coefficient $5$.

The two numbers that satisfy these conditions are $9$ and $-4$, because $9 \times -4 = -36$ and $9 + (-4) = 5$. Rewrite the middle term $5x$ as $9x - 4x$. The trinomial becomes $9x^2 + 9x - 4x - 4$.

Group the terms into two pairs: $(9x^2 + 9x) - (4x + 4)$. Factor out the greatest common factor (GCF) from each group. From $9x^2 + 9x$, the GCF is $9x$, and from $-4x - 4$, the GCF is $-4$. This gives $9x(x + 1) - 4(x + 1)$.

Notice that $x + 1$ is a common factor. Factor it out to get $(9x - 4)(x + 1)$. Thus, the trinomial $9x^2 + 5x - 4$ factors as $(9x - 4)(x + 1)$.

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Factoring Trinomials

Factoring trinomials involves rewriting a quadratic expression in the form ax^2 + bx + c as a product of two binomials. This process requires identifying two numbers that multiply to ac (the product of a and c) and add to b. Understanding this concept is crucial for simplifying expressions and solving equations.

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A trinomial is considered prime if it cannot be factored into the product of two binomials with rational coefficients. Recognizing prime trinomials is essential in factoring, as it helps determine when an expression cannot be simplified further. This concept is important for accurately classifying quadratic expressions.

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The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), provides a method for finding the roots of a quadratic equation. While not directly related to factoring, it can be used to determine if a trinomial can be factored by checking the discriminant (b² - 4ac). If the discriminant is positive, the trinomial can be factored into real numbers.

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