Problem 35 - Introductory Algebra for College Students

Question

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. \(x^{2}+7 x y+6 y^{2}\)

Step-by-Step Solution

Answer

The factorization of the trinomial \(x^{2}+7 x y+6 y^{2}\) is \((x+y)*(x+6y)\).

Step 1: Identify factors

Find factors of 6 which sum up to 7. The pairs of factors of 6 are (1,6) and (2,3). From these pairs, we can observe that (1,6) adds up to 7.

Step 2: Factorize the trinomial

Write \(x^{2}+7 x y+6 y^{2}\) as \(x^{2}+xy+6xy+6 y^{2}\) and factor by grouping. Resulting in \((x+1y)*(x+6y)\). This gives the factorization of the given trinomial as two binomials.

Step 3: Verify using FOIL

FOIL stands for First Outside Inside Last. It means you multiply the first terms of each binomial, then the outside terms, the inside terms, and finally the last terms. So, \((x+y)*(x+6y)\) becomes \(x^{2}+xy+6xy+6y^{2}\), which further simplifies to the original exercise expression \(x^{2}+7 x y+6 y^{2}\). This confirms the factorization is correct.

Key Concepts

FOIL Multiplication

Understanding FOIL multiplication is essential when working with polynomials, specifically when trying to multiply two binomials. FOIL stands for First, Outside, Inside, Last. It describes the order in which you should multiply the terms of two binomials. Here's a simple breakdown of the process:

• First: Multiply the first terms in each binomial.
• Outside: Multiply the outer terms in the product.
• Inside: Multiply the inner terms.
• Last: Multiply the last terms in each binomial.

After performing these steps, you should combine any like terms to simplify the expression. For example, when verifying the factorization of a trinomial such as \(x^{2}+7xy+6y^{2}\), using FOIL on the factors \((x+y)(x+6y)\) will conveniently return you to the original quadratic expression, confirming the correctness of your factorization.

Polynomial Factorization

Polynomial factorization is the process of breaking down a complex polynomial into products of simpler polynomials that, when multiplied together, give back the original polynomial. For trinomials specifically, where the general form is \(ax^2 + bx + c\), we particularly look for two binomials whose product will be the original trinomial.

The step-by-step solution for factorization involves identifying a pair of numbers that both multiply to give the constant term (in this case, 6) and add to give the linear coefficient (in this case, 7). This is a crucial step, as selecting the right pair of numbers will directly influence the success of the factorization. As in our example, the process led to identifying the pair (1,6) and subsequently to the factorization \((x+y)(x+6y)\).

Solving Quadratic Equations

Solving quadratic equations is another critical skill in algebra. These are equations of the form \(ax^2 + bx + c = 0\), and they can be solved using several methods such as factoring, completing the square, using the quadratic formula, or graphing. Factorization is often the quickest method when it is feasible.

The central theme in solving quadratic equations through factoring is to express the quadratic as a product of binomials, set each binomial equal to zero, and solve for the variable. In the exercise example, if we had \(x^2 + 7xy + 6y^2 = 0\), we would use the factors \((x+y)\) and \((x+6y)\) and set up the equations: \(x+y = 0\) and \(x+6y = 0\), thus finding the values of 'x' that satisfy the equation.

Binomial Factorization

Binomial factorization involves breaking down a polynomial expression into two binomial expressions that when multiplied together reproduce the original polynomial. Particularly when dealing with trinomials, finding the correct binomial factors can be a matter of trial and error, guided by the principles of polynomial factorization.

To factor a trinomial, you look for two numbers that add up to the coefficient of the middle term and multiply to the constant term. Once found, these numbers are used to split the middle term and then group and factor by common factors, which results in binomial factors. In our exercise, encountering the trinomial \(x^2 + 7xy + 6y^2\), we can eventually deduce the binomial factors to be \((x+y)\) and \((x+6y)\).