\left(25k^3-30k^2-2k+8\right)\div \ \left(5k-4\right)

To solve the division \( \frac{25k^3 - 30k^2 - 2k + 8}{5k - 4} \), we will use polynomial long division. Here’s the step-by-step process:

### Step 1: Divide the first terms
Take the first term of the dividend \( 25k^3 \) and divide it by the first term of the divisor \( 5k \). This gives:
\[
\frac{25k^3}{5k} = 5k^2
\]
Now, multiply the entire divisor \( 5k - 4 \) by \( 5k^2 \):
\[
5k^2(5k - 4) = 25k^3 - 20k^2
\]

### Step 2: Subtract the result from the original polynomial
Now subtract \( 25k^3 - 20k^2 \) from the original polynomial:
\[
(25k^3 - 30k^2 - 2k + 8) - (25k^3 - 20k^2) = -10k^2 - 2k + 8
\]

### Step 3: Divide the new first terms
Now divide the first term of the new polynomial \( -10k^2 \) by the first term of the divisor \( 5k \):
\[
\frac{-10k^2}{5k} = -2k
\]
Multiply \( -2k \) by \( 5k - 4 \):
\[
-2k(5k - 4) = -10k^2 + 8k
\]

### Step 4: Subtract again
Subtract \( -10k^2 + 8k \) from \( -10k^2 - 2k + 8 \):
\[
(-10k^2 - 2k + 8) - (-10k^2 + 8k) = -10k + 8
\]

### Step 5: Divide the new first terms
Now divide the first term of the new polynomial \( -10k \) by \( 5k \):
\[
\frac{-10k}{5k} = -2
\]
Multiply \( -2 \) by \( 5k - 4 \):
\[
-2(5k - 4) = -10k + 8
\]

### Step 6: Subtract the final time
Subtract \( -10k + 8 \) from \( -10k + 8 \):
\[
(-10k + 8) - (-10k + 8) = 0
\]

### Final Answer
Thus, the quotient is:
\[
5k^2 - 2k - 2
\]

Would you like further details or explanations on any of these steps?

Here are 5 related questions to practice:
1. Divide \( 12x^3 - 16x^2 + 8x - 4 \) by \( 4x - 2 \).
2. Perform polynomial division of \( 15y^3 + 10y^2 - 5y + 1 \) by \( 5y - 1 \).
3. Factor and divide \( 30k^3 - 45k^2 + 15k \) by \( 15k \).
4. What is the remainder when \( 4x^3 + 2x - 1 \) is divided by \( 2x - 1 \)?
5. Use synthetic division to divide \( 8k^3 - 12k^2 + 6k - 4 \) by \( k - 2 \).

**Tip**: Polynomial long division is very similar to numerical long division—focus on dividing the highest degree terms first!

Learn how to solve the polynomial division (25k^3 - 30k^2 - 2k + 8) ÷ (5k - 4) using the polynomial long division method. This step-by-step solution explains dividing complex polynomials, focusing on key algebraic concepts. Ideal 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