Question

which expression is equivalent to (7a^2b + 10a^2b^2 + 14a^2b^3)?
(circ) (ab(7a^2 + 10ab + 14b^2))
(circ) (a^2b(7 + 10b + 14b^2))
(circ) (7a^2(b + 3b^2 + 7b^3))
(circ) (7a^2b^3(b^2 + 3b + 7))

Explanation:

Step1: Find the GCF

The given expression is \(7a^{2}b + 10a^{2}b^{2}+14a^{2}b^{3}\). The greatest common factor (GCF) of the terms \(7a^{2}b\), \(10a^{2}b^{2}\), and \(14a^{2}b^{3}\) is \(a^{2}b\) (since the lowest power of \(a\) is \(a^{2}\), the lowest power of \(b\) is \(b^{1}\), and the GCF of 7, 10, 14 is 1? Wait, no, wait: Wait, 7, 10, 14: GCF of 7,10,14 is 1? Wait no, wait the terms are \(7a^{2}b\), \(10a^{2}b^{2}\), \(14a^{2}b^{3}\). So the GCF for the coefficients 7,10,14 is 1? Wait no, wait maybe I made a mistake. Wait, no, let's check each term:

First term: \(7a^{2}b = a^{2}b\times7\)

Second term: \(10a^{2}b^{2}=a^{2}b\times10b\)

Third term: \(14a^{2}b^{3}=a^{2}b\times14b^{2}\)

Ah, right! So the GCF is \(a^{2}b\).

Step2: Factor out the GCF

Factor out \(a^{2}b\) from each term:

\(7a^{2}b + 10a^{2}b^{2}+14a^{2}b^{3}=a^{2}b(7 + 10b + 14b^{2})\)

Wait, looking at the options:

Option 2: \(a^{2}b(7 + 10b + 14b^{2})\) – wait, the option is written as \(a^{2}b(7 + 10b + 14b^{2})\)? Wait the second option is \(a^{2}b(7 + 10b + 14b^{2})\)? Let me check the options again.

Wait the options are:

1. \(ab(7a^{2}+10ab + 14b^{2})\) – if we expand this, we get \(7a^{3}b + 10a^{2}b^{2}+14ab^{3}\), which is not the original expression.

1. \(a^{2}b(7 + 10b + 14b^{2})\) – expanding this: \(a^{2}b\times7 + a^{2}b\times10b + a^{2}b\times14b^{2}=7a^{2}b + 10a^{2}b^{2}+14a^{2}b^{3}\), which matches the original expression.

1. \(7a^{2}(b + 3b^{2}+7b^{3})\) – expanding: \(7a^{2}b + 21a^{2}b^{2}+49a^{2}b^{3}\), not matching.

1. \(7a^{2}b^{3}(b^{2}+3b + 7)\) – expanding: \(7a^{2}b^{5}+21a^{2}b^{4}+49a^{2}b^{3}\), not matching.

So the correct factoring is factoring out \(a^{2}b\) from each term, resulting in \(a^{2}b(7 + 10b + 14b^{2})\), which is the second option.

Answer:

The correct option is the second one: \(a^{2}b(7 + 10b + 14b^{2})\) (the option with the circle next to \(a^{2}b(7 + 10b + 14b^{2})\))