Problem 33 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ 6 y^{4}-15 y^{3} $$

Step-by-Step Solution

Answer

The expression factors completely as \(3y^3(2y - 5)\).

Step 1: Identify Common Factors

The first step is to identify the greatest common factor (GCF) of the terms in the expression \( 6y^4 - 15y^3 \). Examine the coefficients \(6\) and \(15\), which both have a common factor of \(3\). Additionally, both terms include a factor of \(y^3\). Thus, the GCF is \(3y^3\).

Step 2: Factor Out the GCF

Factor the GCF, \(3y^3\), from the original expression. This involves dividing each term by the GCF: \[ 6y^4 \div 3y^3 = 2y \] \[ -15y^3 \div 3y^3 = -5 \] Thus, the expression becomes: \[ 3y^3(2y - 5) \]

Step 3: Verify the Factored Form

Expand the factored expression \(3y^3(2y - 5)\) to verify correctness. Distribute \(3y^3\) to both terms inside the parentheses: \[ 3y^3 imes 2y = 6y^4 \] \[ 3y^3 imes (-5) = -15y^3 \] The expression expands back to the original \(6y^4 - 15y^3\), confirming the factorization is correct.

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is a key concept when working with algebraic expressions and polynomials. It is the largest factor that divides each term in the expression without leaving a remainder.
To find the GCF, you need to look for both numerical and variable components that are common among the terms.
In the expression \( 6y^4 - 15y^3 \):

• First, focus on the numerical part of the terms. Here, 6 and 15 have a numerical GCF of 3.
• Next, examine the variable part. Both terms include powers of \( y \). The smallest power of \( y \) in the terms is \( y^3 \), which serves as the variable part of the GCF.

Putting it all together, the GCF is \( 3y^3 \). By factoring it out, you simplify the expression to make solving or further manipulation easier.

Polynomials

Polynomials are expressions made up of variables and coefficients, involving addition, subtraction, and non-negative integer exponents of variables. They form the building blocks for many algebraic operations you encounter.
The expression \( 6y^4 - 15y^3 \) is a polynomial composed of two terms. Each term has a coefficient and a variable raised to a power:

• \( 6y^4 \): Coefficient is 6, and variable \( y \) is raised to the 4th power.
• \( -15y^3 \): Coefficient is -15, and variable \( y \) is raised to the 3rd power.

Understanding polynomials is crucial as they often represent various mathematical functions we need to solve or simplify. Factoring polynomials, like in this case, helps in simplifying them and facilitates solving equations or inequalities.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They are fundamental in expressing mathematical ideas and forming equations and formulas.
In \( 6y^4 - 15y^3 \), we handle an algebraic expression with specific components:

• Variables, which are represented by \( y \) in both terms.
• Coefficients, the numerical parts (6 and -15) that multiply the variables.

Factoring is a way to simplify algebraic expressions by breaking them down into smaller parts that are easier to work with. It makes the expressions more manageable and reveals insights into the relationships between the terms. Knowing how to manipulate algebraic expressions forms the groundwork for solving complex algebraic and calculus problems.