https://file.aiquickdraw.com/mathbot/file/17282582582778yjhgh1t.jpg

Let's solve this polynomial long division problem step by step.

The given expression is:
\[
\frac{x^3 + 5x^2 + 7x - 13}{x^2 + 3x - 4}
\]

We are required to divide \(x^3 + 5x^2 + 7x - 13\) by \(x^2 + 3x - 4\) and show that it simplifies to:
\[
x + 2 + \frac{5}{x + 4}
\]

### Step 1: Perform the Long Division

1. **Divide the leading term**: Divide the highest degree term of the numerator \(x^3\) by the highest degree term of the denominator \(x^2\):
   \[
   \frac{x^3}{x^2} = x
   \]
   So, the first term of the quotient is \(x\).

2. **Multiply and subtract**: Multiply \(x\) by the entire denominator \(x^2 + 3x - 4\):
   \[
   x(x^2 + 3x - 4) = x^3 + 3x^2 - 4x
   \]
   Subtract this from the original numerator:
   \[
   (x^3 + 5x^2 + 7x - 13) - (x^3 + 3x^2 - 4x) = 2x^2 + 11x - 13
   \]

3. **Repeat the process**: Divide the new leading term \(2x^2\) by the highest degree term of the denominator \(x^2\):
   \[
   \frac{2x^2}{x^2} = 2
   \]
   Add this to the quotient.

4. **Multiply and subtract again**: Multiply \(2\) by the entire denominator \(x^2 + 3x - 4\):
   \[
   2(x^2 + 3x - 4) = 2x^2 + 6x - 8
   \]
   Subtract this from the current remainder:
   \[
   (2x^2 + 11x - 13) - (2x^2 + 6x - 8) = 5x - 5
   \]

### Step 2: Write the Quotient and the Remainder

The quotient so far is:
\[
x + 2
\]
And the remainder is:
\[
5x - 5
\]

### Step 3: Express the Remainder as a Fraction

We write the remainder over the original denominator:
\[
\frac{5x - 5}{x^2 + 3x - 4}
\]

### Step 4: Simplify the Remainder Fraction

Factor the denominator \(x^2 + 3x - 4\):
\[
x^2 + 3x - 4 = (x + 4)(x - 1)
\]

And factor the numerator \(5x - 5\):
\[
5x - 5 = 5(x - 1)
\]

This allows us to simplify:
\[
\frac{5(x - 1)}{(x + 4)(x - 1)} = \frac{5}{x + 4}
\]

### Final Answer

The division simplifies to:
\[
x + 2 + \frac{5}{x + 4}
\]

This matches the given expression, confirming that the calculation is correct.

Would you like a more detailed explanation or have any questions?

### Follow-up Questions
1. What are the steps to perform polynomial long division?
2. How do you factor quadratic expressions like \(x^2 + 3x - 4\)?
3. Why is it important to express the remainder in the form of a fraction?
4. Can synthetic division be used for non-linear divisors?
5. What happens if the divisor polynomial has complex roots?

### Tip
When performing polynomial long division, always align terms by their degree to avoid errors in subtraction.

Divide x^3 + 5x^2 + 7x - 13 by x^2 + 3x - 4 and hence show that (x^3 + 5x^2 + 7x - 13) / (x^2 + 3x - 4) = x + 2 + 5 / (x + 4).

Learn how to divide polynomials by performing long division of x^3 + 5x^2 + 7x - 13 by x^2 + 3x - 4, and simplify the expression to x + 2 + 5 / (x + 4). This step-by-step guide covers key concepts like polynomial division and factoring, suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation