Problem 272 - Elementary Algebra

Question

Factor. \(48 q^{3}-24 q^{2}+3 q\)

Step-by-Step Solution

Answer

3q(16q^2 - 8q + 1)

Step 1: Identify the Greatest Common Factor (GCF)

First, identify the greatest common factor (GCF) of the coefficients and the variable terms. The coefficients are 48, 24, and 3, with a common factor of 3. Each term also has at least one q. So, the GCF is 3q.

Step 2: Factor out the GCF

Divide each term by the GCF (3q) and write the expression as a product of the GCF and the resulting polynomial. \[48 q^{3} - 24 q^{2} + 3 q = 3 q (16 q^{2} - 8 q + 1)\]

Key Concepts

Greatest Common Factor (GCF)

Understanding the Greatest Common Factor (GCF) is essential when factoring polynomials.
The GCF is the largest number and variable that can evenly divide all terms in a polynomial.
For coefficients, find the largest number that divides each coefficient exactly.
For variables, look for the lowest degree of the variable common to all terms.
In our example, the coefficients are 48, 24, and 3.
The greatest number that divides all three is 3.
Each term also contains at least one 'q', so the GCF for the variables is 'q'.
Combining both, the GCF for the polynomial is 3q.
Factoring out the GCF simplifies the polynomial and is the first step in many algebraic processes.

Algebraic Expressions

Algebraic expressions are combinations of variables, coefficients, and arithmetic operations.
These can include addition, subtraction, multiplication, and division.
The expression from the exercise, 48q³ - 24q² + 3q, has multiple terms.
Each term consists of a coefficient (number) and a variable raised to a power.
Terms in a polynomial are separated by addition or subtraction.
Knowing the structure of algebraic expressions helps identify like terms and apply operations accurately.
Additionally, recognizing how to combine like terms and use the proper order of operations is key to manipulating these expressions effectively.

Polynomial Division

Polynomial division involves dividing each term of the polynomial by a common factor.
After identifying the GCF, divide each term by this factor.
For our polynomial 48q³ - 24q² + 3q, we found the GCF to be 3q.
Now, divide each term:
- For 48q³, dividing by 3q gives 16q².
- For 24q², dividing by 3q gives 8q.
- For 3q, dividing by 3q gives 1.
After division, we write the polynomial as a product of the GCF and the resulting polynomial (16q² - 8q + 1).
The factorization is: 48q³ - 24q² + 3q = 3q (16q² - 8q + 1).
Polynomial division helps simplify expressions and solve polynomial equations.