 (2x⁴ + 3x³ - 5x² + 7x - 1) ÷ (x + 1)


To divide the polynomial \( 2x^4 + 3x^3 - 5x^2 + 7x - 1 \) by \( x + 1 \), we can use **polynomial long division**. Let's work through it step by step:

### Step 1: Set up the division
\[
\frac{2x^4 + 3x^3 - 5x^2 + 7x - 1}{x + 1}
\]

### Step 2: Divide the leading term
We divide the leading term of the dividend \( 2x^4 \) by the leading term of the divisor \( x \):
\[
\frac{2x^4}{x} = 2x^3
\]
This is the first term of the quotient.

### Step 3: Multiply and subtract
Multiply \( 2x^3 \) by \( x + 1 \):
\[
2x^3(x + 1) = 2x^4 + 2x^3
\]
Now subtract this from the original polynomial:
\[
(2x^4 + 3x^3 - 5x^2 + 7x - 1) - (2x^4 + 2x^3) = (3x^3 - 2x^3) - 5x^2 + 7x - 1 = x^3 - 5x^2 + 7x - 1
\]

### Step 4: Divide the next leading term
Now divide \( x^3 \) by \( x \):
\[
\frac{x^3}{x} = x^2
\]
This is the second term of the quotient.

### Step 5: Multiply and subtract
Multiply \( x^2 \) by \( x + 1 \):
\[
x^2(x + 1) = x^3 + x^2
\]
Now subtract this from the current remainder:
\[
(x^3 - 5x^2 + 7x - 1) - (x^3 + x^2) = (-5x^2 - x^2) + 7x - 1 = -6x^2 + 7x - 1
\]

### Step 6: Divide the next leading term
Divide \( -6x^2 \) by \( x \):
\[
\frac{-6x^2}{x} = -6x
\]
This is the third term of the quotient.

### Step 7: Multiply and subtract
Multiply \( -6x \) by \( x + 1 \):
\[
-6x(x + 1) = -6x^2 - 6x
\]
Now subtract this from the current remainder:
\[
(-6x^2 + 7x - 1) - (-6x^2 - 6x) = (7x + 6x) - 1 = 13x - 1
\]

### Step 8: Divide the next leading term
Divide \( 13x \) by \( x \):
\[
\frac{13x}{x} = 13
\]
This is the fourth term of the quotient.

### Step 9: Multiply and subtract
Multiply \( 13 \) by \( x + 1 \):
\[
13(x + 1) = 13x + 13
\]
Now subtract this from the current remainder:
\[
(13x - 1) - (13x + 13) = -1 - 13 = -14
\]

### Step 10: Conclusion
The quotient is \( 2x^3 + x^2 - 6x + 13 \) and the remainder is \( -14 \).

Thus, the result of the division is:
\[
\frac{2x^4 + 3x^3 - 5x^2 + 7x - 1}{x + 1} = 2x^3 + x^2 - 6x + 13 - \frac{14}{x + 1}
\]

Would you like more details or have any questions?

Here are five related questions:
1. How do you apply synthetic division to this problem?
2. Can polynomial division be used to factor polynomials?
3. What are the properties of the remainder when dividing polynomials?
4. How can you verify the result of polynomial division?
5. What happens when the divisor is a higher-degree polynomial?

**Tip**: When dividing polynomials, always align terms by their degrees to avoid mistakes in subtraction.

(2x⁴ + 3x³ - 5x² + 7x - 1) ÷ (x + 1)

Understand how to divide the polynomial 2x⁴ + 3x³ - 5x² + 7x - 1 by x + 1 using polynomial long division. This step-by-step guide walks through the quotient and remainder calculations, applying key algebraic concepts for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation