Problem 9 - College Algebra with Modeling and Visualization

Question

Factor out the greatest common factor:. \(15 x^{3}+10 x^{2}-30 x\)

Step-by-Step Solution

Answer

The greatest common factor is \(5x\), and the factored form is \(5x(3x^2 + 2x - 6)\).

Step 1: Identify the Terms

The given expression is \(15x^3 + 10x^2 - 30x\). It consists of three terms: \(15x^3\), \(10x^2\), and \(-30x\).

Step 2: Determine the Numerical GCF

Find the greatest common factor (GCF) of the numerical coefficients: 15, 10, and 30. In this case, the GCF is 5, since it is the largest number that divides all of them.

Step 3: Determine the Variable GCF

Each term has an \(x\) variable. The smallest power of \(x\) is 1 (from \(-30x\)), so the GCF for the variables is \(x\).

Step 4: Factor Out the GCF

Combine the numerical and variable GCFs. In this case, that is \(5x\). Divide each term of the original expression by \(5x\) to factor it out: - \(\frac{15x^3}{5x} = 3x^2\) - \(\frac{10x^2}{5x} = 2x\) - \(\frac{-30x}{5x} = -6\). Thus, the expression becomes \(5x(3x^2 + 2x - 6)\).

Step 5: Verify the Factorization

Expand \(5x(3x^2 + 2x - 6)\) to check correctness:- \(5x \times 3x^2 = 15x^3\)- \(5x \times 2x = 10x^2\)- \(5x \times -6 = -30x\) The original expression \(15x^3 + 10x^2 - 30x\) is obtained, confirming the factorization is correct.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is a fundamental concept in algebra that helps simplify expressions by finding the largest factor shared by all terms in a set. It is especially useful in factoring, which is key to solving polynomial equations. Imagine you are decluttering a room: the GCF is like finding the largest container that can fit all similar items.

The process begins by identifying the numerical part of the terms. Consider the coefficients: 15, 10, and 30. The GCF here is 5 because it is the largest number that divides each coefficient evenly.

Next, examine the variables. In an expression like ours, terms share a common variable. Check for the smallest power of that variable. Here, the smallest power of \(x\) in \(15x^3, 10x^2,\) and \(-30x\) is \(x^1\).

Combine both the numerical and variable factors to get the final GCF. For this exercise, the GCF is \(5x\). Factoring this out from the expression noticeably simplifies it, making it easier to work with or solve.

Polynomial Expressions

Polynomial expressions are a type of algebraic expression that consist of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. Think of them as mathematical sentences that represent a wide array of possible curves on a graph.

The expression \(15x^3 + 10x^2 - 30x\) is a classic example, known as a trinomial since it has three distinct terms. Each term can be seen as a mini-expression where different parts (the coefficient and the variable with its exponent) combine.

Recognizing a polynomial's structure is crucial:

• Variables: The letters that can represent unknown values (e.g., \(x\)).
• Coefficients: The numerical portion (e.g., 15, 10, -30).
• Exponents: Power raised on the variables (e.g., the 3 in \(x^3\)).

Factoring polynomials often involves finding these common components and expressing them in a simplified form, making solving equations or further manipulation easier.

Algebraic Expression

An algebraic expression is a combination of numbers, operations, and variables. These expressions represent mathematical relationships and can take on a variety of forms.

• Linear terms: Like \(-30x\), where the variable’s power is one.
• Quadratic terms: Like \(10x^2\), featuring a squared variable.
• Cubic terms: Such as \(15x^3\), with variables to the third power.

The ability to manipulate these through operations like addition, subtraction, multiplication, and factoring is the foundation of algebra.

In the exercise, we've used factoring to simplify the expression \(15x^3 + 10x^2 - 30x\) by extracting the GCF. Factoring helps in simplifying these expressions, making them easier to manage within larger algebraic operations or when solving for variable values.