Problem 88 - Introductory Algebra for College Students

Question

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$24 a^{4} b+60 a^{3} b^{2}+150 a^{2} b^{3}$$

Step-by-Step Solution

Answer

The completely factored form of the polynomial \(24 a^{4} b + 60 a^{3} b^{2} + 150 a^{2} b^{3}\) is \(6a^{2}b \cdot (4a^2 + 10ab + 25b^{2})\).

Step 1: Identify Common Factors

The first step to solving this problem is to identify the common factors in each term. In this case, for \(24 a^{4} b, 60 a^{3} b^{2}, 150 a^{2} b^{3}\), the common factors are \(6a^{2}b\). Factorize each term by factoring out \(6a^{2}b\).

Step 2: Factorize

Factorizing out \(6a^{2}b\), we get \[6a^{2}b \cdot (4a^2 + 10ab + 25b^{2}) \]. This is the factored form of the polynomial.

Step 3: Verify the Solution

To verify that the factored form is equivalent to the original polynomial, distribute \(6a^{2}b\) across all terms in the brackets. \[ 6a^{2}b \cdot (4a^2 + 10ab + 25b^{2}) = 24 a^{4} b + 60 a^{3} b^{2} + 150 a^{2} b^{3} \]. This verifies that the solution is correct.

Key Concepts

Common Factors

When dealing with polynomials, identifying common factors is a crucial first step in the factoring process. Common factors are numbers or variables that are shared across all terms in the polynomial.
For the polynomial given:

• Terms: \(24 a^{4} b, 60 a^{3} b^{2}, 150 a^{2} b^{3}\)
• The greatest common factor involves both coefficients and variables.

First, identify the numeric greatest common factor (GCF) of 24, 60, and 150, which is 6. Next, look at the variable parts. Each term has at least \(a^2\) and \(b\) as factors. Therefore, the GCF in terms of variables is \(a^2b\).
Combining these, the overall common factor is \(6a^2b\). Once identified, this common factor can be factored out from the polynomial, simplifying it and preparing it for further factoring or solving.

Polynomials with Multiple Variables

Factoring polynomials with multiple variables involves both numeric and variable components. Consider our polynomial:

• After identifying the common factor \(6a^2b\), it is factored out of each term.
• Start with the coefficients once the GCF is extracted.

Divide each coefficient and variable of the polynomial by the GCF \(6a^2b\). This leaves us with a new polynomial: \[(4a^2 + 10ab + 25b^2)\].
Each term within the parentheses is simplified by the removal of the GCF, keeping the polynomial balanced and ready for further operations or evaluation. This reduction retains the essence of the original polynomial but in a simplified factorized form.

Verification of Factored Form

After factoring a polynomial, it's critical to verify the result to ensure accuracy. This validates that the transformation maintains equivalence with the original polynomial.
To verify, multiply back the factored terms:

• Start with the common factor \(6a^2b\).
• Distribute it across each term within the parentheses.

Multiplying: \(6a^2b \cdot (4a^2 + 10ab + 25b^2)\)

• \(= 24a^4b\)
• \(+ 60a^3b^2\)
• \(+ 150a^2b^3\)

The result matches the original polynomial \(24 a^{4} b + 60 a^{3} b^{2} + 150 a^{2} b^{3}\), verifying that the factoring was done correctly.
This step is vital as it solidifies understanding and confirms the correctness of the process and result.