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Chapter 6: Problem 18

Use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$3 x^{2}-25 x-28$$

Short Answer

Expert verified
The factorization of \(3x^{2} - 25x - 28\) is \((3x - 7)(x + 4)\)

Step by step solution

01

Setting up the equation

Write down the trinomial that is to be factored, \(3x^{2}-25x-28\).
02

Apply AC method

AC method involves multiplying the coefficient of \(x^2\) (A) and the constant (C) in the equation. In this case, A=3 and C=-28, so AC = -84. Now, we need to find two numbers that multiply to AC and add to B (the coefficient of x) which is -25. Those numbers are -7 and 12 because \(-7*12= -84\) and \(-7+12 = -25\) . Now rewrite mid term with these two numbers: \(3x^{2}-7x+12x-28\). Now factor by grouping.
03

Factor by grouping

Rewrite the trinomial as a four-term polynomial and group the terms so that there is a common factor in each group: \(3x^{2}-7x+12x-28 = (3x^{2} -7x) + (12x - 28)\). Factor out the common factor in each group: \(x(3x-7) + 4(3x-7)\). Now, since the terms in parentheses are the same, factor out \(3x-7\) from the entire expression to get \((3x - 7)(x + 4)\).
04

Validate solution by applying FOIL method

Use the FOIL method to expand \((3x - 7)(x + 4)\) which should give the original trinomial. First: \(3x * x = 3x^{2}\), Outer: \(3x * 4 = 12x\), Inner: \(-7 * x = -7x\), Last: \(-7 * 4 = -28\). Adding these together, we get the original trinomial: \(3x^{2} - 7x + 12x - 28 = 3x^{2} - 25x - 28\). Therefore, the factorization is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

AC method
The AC method is a systematic way to factor trinomials, especially useful when the coefficient of the square term is not 1. It involves a few key steps that make it easier to break down complex trinomials into factors. Here鈥檚 how you can go about it:
  • First, identify the coefficients: A is the coefficient of the squared term and C is the constant term. For the trinomial given, \(3x^2 - 25x - 28\), we have \(A = 3\) and \(C = -28\).
  • Next, multiply these two coefficients to find the product AC: \(3 \times -28 = -84\).
  • Now, look for two numbers that multiply to AC and add up to B, the coefficient of the linear term \(-25x\). In this case, \(-7\) and \(12\) fit these criteria because \(-7 \times 12 = -84\) and \(-7 + 12 = -25\).
  • Replace the middle term \(-25x\) with these two numbers to rewrite the trinomial: \(3x^2 - 7x + 12x - 28\).
This prepares the trinomial for the next step, factoring by grouping.
FOIL method
The FOIL method is a technique used to multiply two binomials and is also great for checking your work when factoring trinomials. FOIL stands for:
  • First: Multiply the first terms in each binomial.
  • Outer: Multiply the outer terms in the product.
  • Inner: Multiply the inner terms.
  • Last: Multiply the last terms in each binomial.
Let's apply this to the factorized expression \( (3x - 7)(x + 4) \):- First: \(3x \times x = 3x^2\)- Outer: \(3x \times 4 = 12x\)- Inner: \(-7 \times x = -7x\)- Last: \(-7 \times 4 = -28\)Add these products together to check if they match the original trinomial: \(3x^2 - 25x - 28\). Since it does, you've confirmed your factorization is correct. FOIL not only helps with multiplication but also serves as a useful tool for verification.
factoring by grouping
Factoring by grouping helps break down a polynomial into simpler parts by organizing its structure into pairs of terms. Here's how it's typically done:
  • With our trinomial rewritten as a four-term polynomial \(3x^2 - 7x + 12x - 28\), begin by grouping the terms: \((3x^2 - 7x) + (12x - 28)\).
  • Next, factor out the greatest common factor (GCF) from each group. In the first group \((3x^2 - 7x)\), the GCF is \(x\), yielding \(x(3x - 7)\). For the second group \((12x - 28)\), the GCF is \(4\), resulting in \(4(3x - 7)\).
  • Notice how \((3x - 7)\) appears in both groups. This allows us to factor out \((3x - 7)\) from the entire expression, leading to the factored form \((3x - 7)(x + 4)\).
The grouping method is effective when a polynomial can be reorganized to reveal common factors in pairs, making it a powerful strategy for complex expressions. Combining this method with others like the AC method strengthens your factoring toolbox.

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