Which expression is equivalent to $7 a^2 b+10 a^2 b^2+14 a^2 b^3$?
A. $a b\left(7 a^2+10 a b+14 b^2\right)$
B. $a^2 b\left(7+10 b+14 b^2\right)$
C. $7 a^2\left(b+3 b^2+7 b^3\right)$
D. $7 a^2 b^3\left(b^2+3 b+7\right)$
Answer (1)
• Identify the greatest common factor (GCF) of the terms in the expression, which is a 2 b .
• Factor out the GCF from the expression: 7 a 2 b + 10 a 2 b 2 + 14 a 2 b 3 = a 2 b ( 7 + 10 b + 14 b 2 ) .
• Compare the factored expression with the given options. • The equivalent expression is a 2 b ( 7 + 10 b + 14 b 2 ) .
Explanation
Understanding the Problem We are given the expression 7 a 2 b + 10 a 2 b 2 + 14 a 2 b 3 and asked to find an equivalent expression from the options provided. To do this, we will factor the given expression.
Finding the Greatest Common Factor First, we identify the greatest common factor (GCF) of the terms in the expression. The coefficients are 7, 10, and 14. The greatest common divisor of these numbers is 1, as shown by the calculation tool. The variables are a 2 b , a 2 b 2 , and a 2 b 3 . The GCF of these terms is a 2 b .
Factoring the Expression Now, we factor out the GCF, which is a 2 b , from the expression: 7 a 2 b + 10 a 2 b 2 + 14 a 2 b 3 = a 2 b ( 7 + 10 b + 14 b 2 )
Identifying the Equivalent Expression Finally, we compare the factored expression a 2 b ( 7 + 10 b + 14 b 2 ) with the given options. The matching expression is a 2 b ( 7 + 10 b + 14 b 2 ) .
Examples
Factoring expressions is a fundamental skill in algebra and is used in many real-world applications. For example, if you are designing a rectangular garden and want to determine the dimensions that will give you a certain area, you might need to factor a quadratic expression. Similarly, in physics, factoring can help simplify equations that describe the motion of objects or the behavior of electrical circuits. Understanding how to factor expressions allows you to solve problems more efficiently and gain a deeper understanding of the relationships between different variables.
