Problem 5 - 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. $$2 x^{2}+11 x+12$$

Step-by-Step Solution

Answer

The factorization of \(2x^{2}+11x+12\) is \((x+4)(2x+3)\).

Step 1: Searching for a pair of numbers

Look for two numbers that both add up to 11 and multiply to 24. The numbers 8 and 3 fulfill this requirement.

Step 2: Factorization of the trinomial

Rewrite the middle term of the trinomial as the sum of the products of \(x\) and each of the two numbers found in step 1: \(2x^{2}+8x+3x+12\). This can be factored by grouping, first the first two terms and then the last two: \(2x(x+4)+3(x+4)\). Since the expression \(x+4\) is common, the trinomial can be factored as \((x+4)(2x+3)\).

Step 3: Verification by FOIL

The FOIL method is applied to the factored form to verify it. Multiply First \(x*2x=2x^{2}\), Outer \(x*3=3x\), Inner \(4*2x=8x\), and the Last terms \(4*3=12\) together. Add these terms to get back to the original trinomial: \(2x^{2}+8x+3x+12 = 2x^{2}+11x+12\).

Key Concepts

FOIL Method

The FOIL method is a handy technique used to multiply two binomials. This process helps confirm if the factorization of a trinomial is correct. The term FOIL stands for First, Outer, Inner, and Last, which are the elements you multiply together to simplify binomials.

Here's how it works:

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

Once you have performed these multiplications, you add all the resulting products together to verify the expression. In our given example, after factoring the trinomial to \(2x+3\)(x+4), the FOIL method ensures that when you multiply these factors back out, you retrieve the original trinomial \(2x^2 + 11x + 12\). With practice, mastering the FOIL method can solidify your understanding of binomial multiplication.

Factoring by Grouping

Factoring by grouping is a reliable technique for simplifying polynomials, like the trinomial in our example. It is particularly useful when a trinomial doesn't immediately factor into easily recognizable bins, especially with a leading coefficient.

For our sample trinomial, \(2x^2 + 11x + 12\), the first task is to express the middle term as a combination of two products, consistent with the leading and constant coefficients. We convert 11x into 8x and 3x because:

• 8 and 3 add up to 11, the middle term's coefficient.
• Their product, 24, matches the product of the first and last coefficients in the original trinomial.

After rewriting as \(2x^2 + 8x + 3x + 12\), the trinomial is split into two binomials that can be factored separately. This becomes \(2x(x+4) + 3(x+4)\), illustrating the grouping step. The expression \(x+4\) is the common factor, allowing you to finalize the factorization into \(x+4)(2x+3)\). This approach turns an intimidating expression into a product of simpler ones.

Prime Trinomials

When dealing with trinomials, finding factors is not always possible. A prime trinomial is one that cannot be expressed as a product of two binomials with real coefficients. It's similar to how a prime number has no divisors other than 1 and itself.

To determine if a trinomial is prime, you explore all combinations of its terms to see if any lead to factorization. If all attempts to rewrite the trinomial through methods like factoring by grouping or testing potential factor pairs fail, then the trinomial is declared prime.

In our specific example, the trinomial \(2x^2 + 11x + 12\) was successfully factored, which means it is not prime. However, some trinomials due to their specific coefficients might resist simplifying and remain in their given polynomial form. Knowing when to label a trinomial as prime is crucial in simplifying polynomial expressions efficiently.