- Identify the GCF of the terms \(7a^2b\), \(10a^2b^2\), and \(14a^2b^3\), which is \(a^2b\).
- Divide each term by \(a^2b\) to find the remaining polynomial: \(\frac{7a^2b}{a^2b} = 7\), \(\frac{10a^2b^2}{a^2b} = 10b\), and \(\frac{14a^2b^3}{a^2b} = 14b^2\).
- Combine the GCF with the resulting polynomial to get the factored form \(a^2b(7 + 10b + 14b^2)\).
- Final Answer: \(\boxed{a^2b(7 + 10b + 14b^2)}\)
Explanation
- Analyzing the Expression
The expression to be factored is \(7a^2b + 10a^2b^2 + 14a^2b^3\). To identify the equivalent expression among the given choices, we need to determine the Greatest Common Factor (GCF) of the three terms and factor it out from the original expression. The three terms are \(7a^2b\), \(10a^2b^2\), and \(14a^2b^3\).
- Finding the Greatest Common Factor
Next, we look for the common factors within the numerical coefficients and the variables.
- Coefficients: The numerical parts of the terms are 7, 10, and 14. Their only common factor is 1.
- Variable 'a': All three terms contain \(a^2\), so \(a^2\) is a common factor.
- Variable 'b': The powers of 'b' present in the terms are \(b^1\), \(b^2\), and \(b^3\). The greatest common power of 'b' present in every term is \(b^1\), or simply \(b\).
Combining these, the GCF of the three terms is \(1 \cdot a^2 \cdot b = a^2b\).
- Factoring the Expression
Now, we factor out the common term \(a^2b\) from the original expression:
\[7a^2b + 10a^2b^2 + 14a^2b^3 = a^2b(7) + a^2b(10b) + a^2b(14b^2)\]
Factoring out \(a^2b\), we get:
\[a^2b(7 + 10b + 14b^2)\]
Comparing this result with the provided options, we see that it matches the second option.
Examples
Factoring is a fundamental skill used in simplifying algebraic expressions, which often arises in physics and engineering when simplifying formulas or solving equations. For example, if you are calculating the total area of three rectangular sections with dimensions \((a^2b \times 7)\), \((a^2b \times 10b)\), and \((a^2b \times 14b^2)\), you can quickly find the combined area by factoring out the common \(a^2b\) factor, simplifying the calculation to \(a^2b(7 + 10b + 14b^2)\). This technique is essential for efficient data analysis and modeling.