Problem 35 - Elementary Algebra

Question

Factor the greatest common factor from each polynomial. \(5 x^{3}-15 x^{2}+20 x\)

Step-by-Step Solution

Answer

The factored form is 5x (x^{2} - 3x + 4)

Step 1: Identify the terms

The polynomial given is 5x^{3}-15x^{2}+20x. Identify each term in the polynomial: 5x^{3}, -15x^{2}, and 20x.

Step 2: Find the GCF of the coefficients

The coefficients of the terms are 5, -15, and 20. Find the greatest common factor (GCF) of these coefficients.Factorize the numbers:5 = 5-15 = -1 * 3 * 520 = 2^2 * 5The GCF of 5, -15, and 20 is 5.

Step 3: Find the GCF of the variables

The variables are x^{3}, x^{2}, and x. The common variable with the smallest power is x.Thus, the GCF of the variables is x.

Step 4: Combine the GCF of coefficients and variables

The GCF of the coefficients is 5, and the GCF of the variables is x. Thus, the overall GCF is 5x.

Step 5: Factor out the GCF

Factor 5x from each term in the polynomial:5x(5x^{3}/5x - 15x^{2}/5x + 20x/5x)Simplify the terms inside the parentheses:5x (x^{2} - 3x + 4)

Step 6: Write the final factored form

The factored form of the polynomial is:5x (x^{2} - 3x + 4)

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest number that can evenly divide all the given numbers. For polynomials, we find the GCF of both coefficients and variables.
The process of finding the GCF involves two main steps:

• Identify the coefficients in the polynomial terms.
• Find the GCF of these coefficients by factoring each number and identifying the highest common factor.

In our example, the coefficients are 5, -15, and 20.
Upon factoring, we see:
5 = 5
-15 = -1 * 3 * 5
20 = 2^2 * 5

The GCF for these coefficients is clearly 5 since it's common in all terms.
This step is crucial because it simplifies the polynomial significantly.

Coefficients

Coefficients are the numerical part of the terms in a polynomial.
Each term in a polynomial is usually formed by a coefficient multiplied by a variable raised to a power.
In our polynomial example 5x^{3}-15x^{2}+20x, the coefficients are 5, -15, and 20.

• The coefficient of 5x^{3} is 5.
• The coefficient of -15x^{2} is -15.
• The coefficient of 20x is 20.

Understanding and identifying these coefficients allows us to find the GCF as part of the factoring process.
Always factorize the coefficients to determine their GCF.

Variables

Variables are symbols used to represent unknown values in polynomials.
They are usually denoted by letters such as x, y, z, etc.
In the polynomial 5x^{3}-15x^{2}+20x, the variable is x.
To find the GCF of the variable part, we look at the smallest power of x across all terms.

• In 5x^{3}, the variable is x raised to the power 3.
• In -15x^{2}, the variable is x raised to the power 2.
• In 20x, the variable is x raised to the power 1.

The smallest power of x is 1, thus the GCF of the variables is x.
This step is crucial in breaking down the polynomial into a simpler form.

Polynomials

Polynomials are algebraic expressions consisting of terms involving variables, coefficients, and exponents.
They can have multiple terms separated by addition or subtraction.
For example, the given polynomial is 5x^{3}-15x^{2}+20x.
Each term in this polynomial has an individual coefficient and a variable raised to a power.

• A term can be something like 5x^{3}.
• The polynomial degree (based on the highest power of x) is 3.
• Properly factoring polynomials makes solving equations and graphing much easier.

Factoring involves finding the GCF and separating it from the polynomial.
This breakdown into parts that are easier to manage helps in further algebraic manipulations.