solve (3x^4-5x^3+4x-6)(x^2-3x+5)^-1 by long division

We need to solve the expression \( \frac{3x^4 - 5x^3 + 4x - 6}{x^2 - 3x + 5} \) using **long division**.

Let's break down the steps:

### Step 1: Set up the division
We want to divide:

\[
\text{Dividend} = 3x^4 - 5x^3 + 0x^2 + 4x - 6
\]
by
\[
\text{Divisor} = x^2 - 3x + 5.
\]

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

\[
\frac{3x^4}{x^2} = 3x^2.
\]

Now, multiply \( 3x^2 \) by the entire divisor \( x^2 - 3x + 5 \):

\[
3x^2(x^2 - 3x + 5) = 3x^4 - 9x^3 + 15x^2.
\]

Now, subtract this from the original polynomial:

\[
(3x^4 - 5x^3 + 0x^2 + 4x - 6) - (3x^4 - 9x^3 + 15x^2) = (0x^4) + (4x^3 - 15x^2 + 4x - 6).
\]

### Step 3: Perform the second division
Now, divide the first term of the new polynomial \( 4x^3 \) by \( x^2 \):

\[
\frac{4x^3}{x^2} = 4x.
\]

Multiply \( 4x \) by the divisor \( x^2 - 3x + 5 \):

\[
4x(x^2 - 3x + 5) = 4x^3 - 12x^2 + 20x.
\]

Subtract this from the current polynomial:

\[
(4x^3 - 15x^2 + 4x - 6) - (4x^3 - 12x^2 + 20x) = 0x^3 + (-3x^2 - 16x - 6).
\]

### Step 4: Perform the third division
Now, divide the first term of the new polynomial \( -3x^2 \) by \( x^2 \):

\[
\frac{-3x^2}{x^2} = -3.
\]

Multiply \( -3 \) by the divisor \( x^2 - 3x + 5 \):

\[
-3(x^2 - 3x + 5) = -3x^2 + 9x - 15.
\]

Subtract this from the current polynomial:

\[
(-3x^2 - 16x - 6) - (-3x^2 + 9x - 15) = 0x^2 + (-25x + 9).
\]

### Step 5: Final result
Now, the quotient is:

\[
3x^2 + 4x - 3,
\]

and the remainder is:

\[
-25x + 9.
\]

Thus, the division can be written as:

\[
\frac{3x^4 - 5x^3 + 4x - 6}{x^2 - 3x + 5} = 3x^2 + 4x - 3 + \frac{-25x + 9}{x^2 - 3x + 5}.
\]

### Summary
The result of the long division is:

\[
3x^2 + 4x - 3 + \frac{-25x + 9}{x^2 - 3x + 5}.
\]

Would you like further details or have any questions?

### Related questions:
1. How do you handle synthetic division for polynomials?
2. Can we perform polynomial division when the degrees of numerator and denominator are equal?
3. What are the advantages of using long division over synthetic division?
4. How do you factor higher-degree polynomials like quartics?
5. Can this method be applied to non-polynomial functions, like rational functions with roots?

### Tip:
Always check for factoring possibilities before performing polynomial division. Factoring can simplify the problem significantly.

Learn how to solve the polynomial division problem (3x^4 - 5x^3 + 4x - 6) / (x^2 - 3x + 5) using long division. This step-by-step guide covers key concepts in polynomial long division and algebra. Suitable for high school 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