Find the quotient and remainder using long division for: x^2+7x+13/x+3

We will perform long division to divide \( x^2 + 7x + 13 \) by \( x + 3 \). 

### Step 1: Set up the division

We write it in long division format:

\[
\frac{x^2 + 7x + 13}{x + 3}
\]

### Step 2: Divide the first term of the dividend by the first term of the divisor

The first term in the dividend is \( x^2 \), and the first term in the divisor is \( x \). Divide \( x^2 \) by \( x \), which gives \( x \).

Now multiply \( x \) by the divisor \( x + 3 \), which gives:
\[
x(x + 3) = x^2 + 3x
\]

### Step 3: Subtract the result from the dividend

Subtract \( x^2 + 3x \) from \( x^2 + 7x + 13 \):
\[
(x^2 + 7x + 13) - (x^2 + 3x) = 4x + 13
\]

### Step 4: Divide the first term of the new dividend by the first term of the divisor

Now divide \( 4x \) by \( x \), which gives \( 4 \).

Multiply \( 4 \) by \( x + 3 \), which gives:
\[
4(x + 3) = 4x + 12
\]

### Step 5: Subtract again

Subtract \( 4x + 12 \) from \( 4x + 13 \):
\[
(4x + 13) - (4x + 12) = 1
\]

### Step 6: Write the quotient and remainder

The quotient is \( x + 4 \), and the remainder is \( 1 \).

So, the result is:
\[
\text{Quotient} = x + 4, \quad \text{Remainder} = 1
\]

Would you like more details or have any questions?

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Here are 5 related questions to further explore:
1. How do you perform synthetic division for \( x^2 + 7x + 13 \) divided by \( x + 3 \)?
2. What happens if you divide polynomials with a degree of zero remainder?
3. Can long division be used with higher degree polynomials?
4. How does the remainder theorem apply to polynomial division?
5. How can we verify the result of long division by multiplying back?

**Tip:** To check your division, always multiply the quotient by the divisor and add the remainder; this should give you the original polynomial.

Find the quotient and remainder using long division for: x^2 + 7x + 13 / x + 3

Learn how to find the quotient and remainder using long division for the polynomial division of x^2 + 7x + 13 by x + 3. This step-by-step solution covers key algebra concepts like long division of polynomials and the remainder theorem. Suitable for students in Grades 9-11.

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