Problem 38 - Introductory Algebra for College Students

Question

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. \(x^{2}-9 x y+14 y^{2}\)

Step-by-Step Solution

Answer

The factorization of the given trinomial \(x^{2}-9 x y+14 y^{2}\) is (x-2y)(x-7y)

Step 1: Identifying the Trinomial

The trinomial provided is \(x^{2}-9 x y+14 y^{2}\). The aim is to factor this expression.

Step 2: Factoring the Trinomial

We must factor this trinomial by finding two numbers that add up to -9 (the coefficient of the middle term) and multiply to 14 (the constant term). The numbers -2 and -7 fit these conditions, as -2 + -7 equals to -9, and -2 multiplied by -7 equals to 14. Therefore, the factorization is \(x^{2}-9 x y+14 y^{2} = (x-2y)(x-7y)\)

Step 3: Checking the Factorization using FOIL

The FOIL method stands for 'First, Outer, Inner, Last' and it is a way for checking the product of two binomials. Applying FOIL to (x-2y)(x-7y), we get: 'First' term: x*x = \(x^{2}\), 'Outer' term: x*-7y = -7xy, 'Inner' term: -2y*x = -2xy, 'Last' term: -2y*-7y = 14\(y^{2}\). Sum these up, we get \(x^{2}\) -9xy + 14\(y^{2}\), which matches the given trinomial, the factorization is correct.

Key Concepts

FOIL Method

The FOIL method is an acronym for First, Outer, Inner, Last, which represents a technique to multiply two binomials. It's a valuable tool to check your factoring of polynomials, ensuring that you've factored them correctly.

Here's how you can employ the FOIL method:

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outermost terms in the product.
• Inner: Multiply the innermost terms.
• Last: Multiply the last terms in each binomial.

Finally, add all the products together to get the expanded polynomial. When using the FOIL method for our example \(x^2-9xy+14y^2\), we multiply \(x-2y\) and \(x-7y\) to verify the factorization.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial expression into a product of simpler factors. It simplifies many algebraic procedures, such as solving equations or graphing functions. A non-prime polynomial can often be expressed as a product of two or more non-constant polynomials.

To factor the given polynomial \(x^2-9xy+14y^2\), we look for two numbers whose product is equal to the constant term (14) and whose sum equals the coefficient of the middle term (-9). Once we determine these numbers are -2 and -7, we can write the trinomial as a product of binomials: \(x-2y\) and \(x-7y\).

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. When an algebraic expression takes the form of a polynomial with three terms, it's called a trinomial. The trinomial from our example, \(x^2-9xy+14y^2\), consists of three terms: \(x^2\), \(9xy\), and \(14y^2\), structured with different powers of x and y and coefficients.

The mastery of simplifying and manipulating algebraic expressions is essential for solving more complex algebra problems. Factoring trinomials is a fundamental skill in this process, which helps in numerous mathematical contexts, such as calculus and differential equations.

Trinomial Factorization

Trinomial factorization is the process of decomposing a trinomial into a product of binomials. To factor a trinomial like \(x^2-9xy+14y^2\), one generally looks for two binomials whose product will yield the original trinomial. The coefficients of the x terms in the binomials must add up to the trinomial's middle term coefficient (-9 in the example), and the constant terms in the binomials must multiply to the trinomial's constant (14 in the example).

Following the steps from our exercise, we identified the correct factors and confirmed their validity with the FOIL method, resulting in the original trinomial expression. Recognizing patterns and practicing with various trinomials enhances this crucial skill in algebra.