Question
5a^2b^2 + 10ab + 25a
Answers
Answered by
GPT-4o mini
To factor the expression \( 5a^2b^2 + 10ab + 25a \), we can look for a common factor in all the terms.
1. First, notice that each term has a common factor of \( 5a \):
- The first term \( 5a^2b^2 \) can be rewritten as \( 5a \cdot ab^2 \).
- The second term \( 10ab \) can be rewritten as \( 5a \cdot 2b \).
- The third term \( 25a \) can be rewritten as \( 5a \cdot 5 \).
2. Now we can factor out \( 5a \):
\[
5a(a b^2 + 2b + 5)
\]
So the expression \( 5a^2b^2 + 10ab + 25a \) factors to:
\[
5a(ab^2 + 2b + 5)
\]
Next, we can check if the quadratic \( ab^2 + 2b + 5 \) can be factored further. The discriminant of this quadratic is given by:
\[
D = b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot 5 = 4 - 20 = -16.
\]
Since the discriminant is negative, \( ab^2 + 2b + 5 \) does not factor further over the real numbers.
Thus, the final answer is:
\[
5a(ab^2 + 2b + 5)
\]
1. First, notice that each term has a common factor of \( 5a \):
- The first term \( 5a^2b^2 \) can be rewritten as \( 5a \cdot ab^2 \).
- The second term \( 10ab \) can be rewritten as \( 5a \cdot 2b \).
- The third term \( 25a \) can be rewritten as \( 5a \cdot 5 \).
2. Now we can factor out \( 5a \):
\[
5a(a b^2 + 2b + 5)
\]
So the expression \( 5a^2b^2 + 10ab + 25a \) factors to:
\[
5a(ab^2 + 2b + 5)
\]
Next, we can check if the quadratic \( ab^2 + 2b + 5 \) can be factored further. The discriminant of this quadratic is given by:
\[
D = b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot 5 = 4 - 20 = -16.
\]
Since the discriminant is negative, \( ab^2 + 2b + 5 \) does not factor further over the real numbers.
Thus, the final answer is:
\[
5a(ab^2 + 2b + 5)
\]