Problem 24 - Introductory Algebra for College Students

Question

Use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$3 x^{2}+22 x-16$$

Step-by-Step Solution

Answer

The factorized form of the trinomial \(3 x^{2}+22 x-16\) is \((3x+8)(x-2)\), and using FOIL method verifies the correct factorization.

Step 1: Factorising the trinomial

This trinomial can be factored using the 'ac' method, which stands for multiplying 'a' and 'c' in the standard form of a trinomial \(ax^{2} + bx + c\). Here a=3, b=22, and c=-16. Multiply 'a' and 'c' (ac) gives -48. Now, we need to find two numbers that multiply to -48 and at the same time add up to 22. The numbers satisfying this condition are 24 and -2. So we write middle term as two terms. We have \(3x^2 + 24x -2x -16\).

Step 2: Grouping the terms

Now form groups of two terms. We get two groups, \((3x^2+24x)\) and \(-2x-16\). Divide the first group by the greatest common factor of the terms in that group and do the same for the second group, it gives us \(x(3x+8)-2(3x+8)\).

Step 3: Factoring common terms

Now, factor out the term \((3x + 8)\) which is common in both groups. This gives us \((3x+8)(x-2)\).

Step 4: Verify using FOIL

To check if we factored correctly, we use the FOIL method to multiply out our binomial. FOIL stands for First, Outer, Inner, Last - reflecting how to position the terms of the binomial. This gives us: First: \(3x \cdot x = 3x^{2}\), Outer: \(3x \cdot -2 = -6x\), Inner: \(8 \cdot x = 8x\), Last: \(8 \cdot -2 = -16\). Sum these terms to get \(3x^{2}+2x-16\) which verifies our factorization is correct. Take note that -6x and 8x combine to give 2x.

Key Concepts

FOIL Method

Understanding the FOIL method is essential when working with polynomials, especially when multiplying binomials or checking the result of a factorization. FOIL is an acronym representing four steps: First, Outer, Inner, Last. This method provides a systematic approach to multiplying two binomials and ensures that all terms are accounted for.

Let's break it down:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms of the binomials.
• Inner: Multiply the inner terms of the binomials.
• Last: Multiply the last terms of each binomial.

Once you’ve multiplied each pair, sum all the products to obtain the final polynomial. If you’re checking a factorized expression, you’ll know you’ve done it correctly if your FOIL result matches the original trinomial.

AC Method Factoring

The 'ac' method of factoring is a useful technique for factoring trinomials, particularly when 'a', the coefficient in front of the quadratic term, is not equal to one. In this method, we start by multiplying the 'a' and 'c' terms from the standard form of a trinomial, which is expressed as \(ax^2 + bx + c\).

The key steps in the 'ac' method are to:

• Find two numbers that multiply to the product of 'a' and 'c' and sum to 'b'.
• Split the middle term, 'bx', into two terms using the numbers found.
• Group the terms into pairs.
• Factor out the greatest common factor from each pair.
• Factor out the common binomial factor from the resulting expression.

Through these steps, we transform the original trinomial into a product of two binomials, revealing its factors.

Common Factoring

Common factoring, also known as factoring by grouping, is a foundational concept in algebra that involves extracting the Greatest Common Factor (GCF) from each term within a polynomial. This process simplifies expressions and is often the first step in various factoring techniques.

To factor by grouping:

• Identify the GCF in a set of terms.
• Divide each term by the GCF.
• Write the original expression as a product of the GCF and the simplified terms.

This process may also involve rearranging terms to identify common factors in a grouping strategy. Doing this simplifies the polynomial and often uncovers a common binomial factor that leads to full factoring of the expression.

Verifying Factorization

After factoring a trinomial or any other polynomial, it's important to verify that the factorization is correct. Verification ensures that the factors, when multiplied out, reproduce the original polynomial without error. The FOIL method, described earlier, is a common verification technique for binomials.

To verify factorization:

• Use the factored form to perform multiplication.
• Expand the product to combine like terms.
• Compare the expanded form with the original polynomial.

If the original polynomial and the expanded form match exactly, the factorization is correct. This verification step is crucial in confirming the accuracy of your work and solidifying your understanding of factoring polynomials.