Problem 50 - 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. $$ 3 x^{2} y-9 x y+45 y $$

Step-by-Step Solution

Answer

The fully factored form is \(3y(x^2 - 3x + 15)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, identify the greatest common factor (GCF) of the terms in the trinomial. The terms are \(3x^2y\), \(-9xy\), and \(45y\). All terms have a common factor of \(3y\), since 3 is a factor of all numerical coefficients, and each term includes \(y\).

Step 2: Factor Out the GCF

Factor out the \(3y\) from each term in the trinomial. This gives: \[3y(x^2 - 3x + 15)\].

Step 3: Check for Further Factoring

Look at the trinomial \(x^2 - 3x + 15\) to see if it can be factored further. Since the discriminant \(b^2 - 4ac\) (where \(a = 1\), \(b = -3\), \(c = 15\)) is negative (\(-3^2 - 4 \times 1 \times 15 = -51\)), it does not factor over the integers.

Key Concepts

Greatest Common Factor

When tackling problems that involve factoring trinomials, the first step is usually to find the Greatest Common Factor (GCF). The GCF is the highest number or expression that divides exactly into all the terms. Identifying it efficiently simplifies the expression.

Let's say you have a trinomial like 3x²y - 9xy + 45y. Begin by examining the coefficients (3, 9, and 45) and finding their common factor. Notice that all coefficients are divisible by 3. This makes 3 part of the GCF.

• The first term includes a y, the second also includes a y, and the third term has a y as well. Thus, y is a common factor.

Combining these, the GCF is 3y. By factoring this out from each term, you simplify the expression, making further analysis of the trinomial easier.

Trinomial Factoring

After factoring out the greatest common factor, which is 3y in the given trinomial, we are left with the expression (x² - 3x + 15). Ideally, the next step would be to factor the resulting trinomial further if possible.

Trinomial factoring involves breaking down a quadratic expression like ax² + bx + c into products of binomials. However, sometimes these trinomials cannot be factored over the integers. The discriminant test (b² - 4ac) helps confirm this.

• For the expression x² - 3x + 15, calculate the discriminant using b = -3, a = 1, and c = 15.
• This results in -3² - 4 × 1 × 15 = -51.
• The negative result indicates that the trinomial cannot be factored further into integer components.

Understanding when further factoring is not possible helps avoid unnecessary computations.

Algebraic Expressions

Algebraic expressions, like the trinomial 3x²y - 9xy + 45y, are combinations of numbers, variables, and operations. They represent a part of algebra that is essential for solving many mathematical problems.

Expressions can be simple, like a lone number or variable, or more complex like the one in this example. Understanding how to manipulate these expressions by processes such as factoring is an invaluable skill.

• Factoring algebraic expressions involves simplifying them into products of factors.
• This can help in simplifying equations and finding solutions more easily.
• It's essential to remember certain operations, like distributing or factoring, to grasp this concept deeply.

In everyday algebra, expressions are the foundations upon which equations and inequalities are built. By mastering factoring and similar processes, the path to higher algebra becomes more navigable.