Certainly! Let's go through the process of dividing the polynomial [tex]\(a^2 + 6a + 5\)[/tex] by [tex]\(a + 1\)[/tex] step-by-step.
### Step-by-Step Solution:
1. Set up the polynomial division:
[tex]\[
\frac{a^2 + 6a + 5}{a + 1}
\][/tex]
2. Divide the first term of the numerator by the first term of the denominator:
- Divide [tex]\(a^2\)[/tex] by [tex]\(a\)[/tex]:
[tex]\[
\frac{a^2}{a} = a
\][/tex]
So, the first term of our quotient is [tex]\(a\)[/tex].
3. Multiply the entire denominator by this first term of the quotient:
- Multiply [tex]\(a\)[/tex] by [tex]\(a+1\)[/tex]:
[tex]\[
a \cdot (a + 1) = a^2 + a
\][/tex]
4. Subtract this result from the original polynomial:
[tex]\[
(a^2 + 6a + 5) - (a^2 + a)
\][/tex]
Breaking it down:
[tex]\[
a^2 + 6a + 5 - a^2 - a
\][/tex]
Simplifying this, we get:
[tex]\[
5a + 5
\][/tex]
5. Repeat the process with the new polynomial [tex]\(5a + 5\)[/tex]:
- Divide [tex]\(5a\)[/tex] by [tex]\(a\)[/tex]:
[tex]\[
\frac{5a}{a} = 5
\][/tex]
So, the next term of our quotient is [tex]\(5\)[/tex].
6. Multiply the entire denominator by this new term of the quotient:
- Multiply [tex]\(5\)[/tex] by [tex]\(a + 1\)[/tex]:
[tex]\[
5 \cdot (a + 1) = 5a + 5
\][/tex]
7. Subtract this result from the remaining polynomial:
[tex]\[
(5a + 5) - (5a + 5)
\][/tex]
This simplifies to:
[tex]\[
0
\][/tex]
8. Combine the quotient and the remainder:
The quotient we have obtained by dividing [tex]\(a^2 + 6a + 5\)[/tex] by [tex]\(a + 1\)[/tex] is [tex]\(a + 5\)[/tex], and the remainder is [tex]\(0\)[/tex].
### Final Answer:
The result of dividing [tex]\(a^2 + 6a + 5\)[/tex] by [tex]\(a + 1\)[/tex] is:
[tex]\[
\boxed{a + 5}
\][/tex]
With a remainder of:
[tex]\[
\boxed{0}
\][/tex]
