Find the quotient and the remainder (if any), when :
`a^(3) - 5a^(2) + 8a + 15` is divided by a + 1.

Answer

To find the quotient and the remainder when dividing the polynomial \( a^3 - 5a^2 + 8a + 15 \) by \( a + 1 \), we can use polynomial long division. Here’s a step-by-step solution:

### Step 1: Set up the division
We are dividing \( a^3 - 5a^2 + 8a + 15 \) by \( a + 1 \).

### Step 2: Divide the leading terms
Divide the leading term of the dividend \( a^3 \) by the leading term of the divisor \( a \):
\[ \frac{a^3}{a} = a^2 \]
This gives us the first term of the quotient.

### Step 3: Multiply and subtract
Multiply \( a^2 \) by the entire divisor \( a + 1 \):
\[ a^2(a + 1) = a^3 + a^2 \]
Now subtract this from the original polynomial:
\[ (a^3 - 5a^2 + 8a + 15) - (a^3 + a^2) = -6a^2 + 8a + 15 \]

### Step 4: Repeat the process
Now, divide the leading term of the new polynomial \( -6a^2 \) by the leading term of the divisor \( a \):
\[ \frac{-6a^2}{a} = -6a \]
This gives us the next term of the quotient.

### Step 5: Multiply and subtract again
Multiply \( -6a \) by the entire divisor \( a + 1 \):
\[ -6a(a + 1) = -6a^2 - 6a \]
Now subtract this from the current polynomial:
\[ (-6a^2 + 8a + 15) - (-6a^2 - 6a) = 14a + 15 \]

### Step 6: Continue the process
Now, divide the leading term \( 14a \) by the leading term of the divisor \( a \):
\[ \frac{14a}{a} = 14 \]
This gives us the next term of the quotient.

### Step 7: Multiply and subtract one last time
Multiply \( 14 \) by the entire divisor \( a + 1 \):
\[ 14(a + 1) = 14a + 14 \]
Now subtract this from the current polynomial:
\[ (14a + 15) - (14a + 14) = 1 \]

### Conclusion
At this point, we have completed the division. The quotient is:
\[ a^2 - 6a + 14 \]
And the remainder is:
\[ 1 \]

### Final Answer
Thus, when \( a^3 - 5a^2 + 8a + 15 \) is divided by \( a + 1 \), the quotient is \( a^2 - 6a + 14 \) and the remainder is \( 1 \).

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