Problem 67 - Algebra A Combined Function

Question

Factor each trinomial completely. Some of these trinomials contain a greatest common factor (other than 1). Don't forget to factor out the GCF first. $$ \begin{aligned} &\frac{1}{2} y^{2}-\frac{9}{2} y-11\\\ &\text { (Factor out } \frac{1}{2} \text { first.) } \end{aligned} $$

Step-by-Step Solution

Answer

The trinomial factors to \(\frac{1}{2}(y - 11)(y + 2)\).

Step 1: Identify the Greatest Common Factor (GCF)

The given trinomial is \( \frac{1}{2}y^2 - \frac{9}{2}y - 11 \). The coefficients are \( \frac{1}{2}, \frac{9}{2}, \text{and } 11 \). The GCF here is \( \frac{1}{2} \).

Step 2: Factor Out the GCF

Factor \( \frac{1}{2} \) out of the trinomial: \[ \frac{1}{2} \left( y^2 - 9y - 22 \right).\]Now work with the expression inside the parentheses: \( y^2 - 9y - 22 \).

Step 3: Factor the Quadratic Expression

Consider \( y^2 - 9y - 22 \). We need two numbers that multiply to \(-22\) and add to \(-9\). These numbers are \(-11\) and \(2\).

Step 4: Express the Quadratic as a Product of Binomials

Using the numbers \(-11\) and \(2\), factor the quadratic: \[ y^2 - 9y - 22 = (y - 11)(y + 2).\]

Step 5: Combine the Factored Forms

Combine the factored form from Step 2 with the factored quadratic:\[\frac{1}{2}(y - 11)(y + 2).\]

Key Concepts

Greatest Common Factor

To begin factoring a trinomial, it is crucial to identify the Greatest Common Factor (GCF) of the terms. The GCF is the largest factor shared by all the terms in an expression.
In the given problem, the trinomial is \( \frac{1}{2} y^{2}-\frac{9}{2} y-11 \). To find its GCF, we look at the coefficients: \( \frac{1}{2}, -\frac{9}{2}, \) and \( -11 \). The common factor here is \( \frac{1}{2} \) because this factor is present in the first two terms, and the whole expression is also divisible by it. By factoring out \( \frac{1}{2} \), the trinomial simplifies, making it easier to manage.
Always remember to search for the GCF before attempting to factor the quadratic part of the trinomial. It's the first step in smoothing the path to complete factoring.

Quadratic Expression

A quadratic expression is a polynomial of the form \( ax^2 + bx + c \). It's called quadratic because the highest power of the variable is two. In our example, once we have factored out the GCF, we work with \( y^2 - 9y - 22 \).
Understanding the structure of a quadratic expression helps in determining the method for factorization. The expression can often be factored further into a product of two binomials if there are numbers that can be multiplied to get the constant term \( c \) and added to result in the middle term \( b \).
Learning to recognize and factor quadratic expressions is a key algebraic skill which provides the foundation for solving more complex equations.

Binomial Product

The process of expressing the quadratic expression as a product of two binomials involves identifying numbers that multiply and add to specific values. For the quadratic \( y^2 - 9y - 22 \), we need two numbers that multiply to \(-22\) (the constant term) and add up to \(-9\) (the coefficient of the middle term), which are \(-11\) and \(2\).
The expression can now be rewritten using these numbers into binomials: \( (y - 11)(y + 2) \). By expressing it as a binomial product, you have effectively factored the quadratic.
Recognizing patterns and leveraging known multiplication facts are vital in quickly finding these crucial numbers.

Algebraic Expressions

The challenge with algebraic expressions often lies in their complexity. These are mathematical phrases that can include numbers, variables, and operations. Factorization simplifies these expressions.
In the larger scope, factoring is often about turning a complex algebraic expression into a simpler product of numbers and other expressions. Consider our initial trinomial: first we factor out the GCF and then express any remaining quadratic expressions as a product of binomials.

• This breakdown makes the expressions more manageable, facilitating easier manipulation or further calculations such as solving equations.
• Algebraic expressions appear in many forms in higher math, and the principles of simplifying them through techniques like factoring are widely applicable.

Recognizing, practicing, and perfecting techniques for working with algebraic expressions prepares students to tackle a broad range of mathematical problems.