(3x²-x²-8x+5) ÷ (x + 2) divide using polynomial long division

To divide \((3x² - x² - 8x + 5)\) by \((x + 2)\) using polynomial long division, follow these steps:

### Step 1: Simplify the Dividend
First, simplify the polynomial in the dividend:
\[
3x^2 - x^2 - 8x + 5 = 2x^2 - 8x + 5
\]
So, the problem now is to divide \(2x^2 - 8x + 5\) by \(x + 2\).

### Step 2: Set Up the Long Division
Set up the long division as follows:

\[
\begin{array}{r|l}
x + 2 & 2x^2 - 8x + 5 \\
\end{array}
\]

### Step 3: Divide the First Term
Divide the first term of the dividend by the first term of the divisor:
\[
\frac{2x^2}{x} = 2x
\]
Write \(2x\) above the division bar.

### Step 4: Multiply and Subtract
Multiply \(2x\) by the entire divisor \((x + 2)\) and subtract from the dividend:
\[
(2x)(x + 2) = 2x^2 + 4x
\]
Subtract this from the dividend:
\[
(2x^2 - 8x + 5) - (2x^2 + 4x) = -12x + 5
\]

### Step 5: Divide the Next Term
Now, divide the next term by the first term of the divisor:
\[
\frac{-12x}{x} = -12
\]
Write \(-12\) above the division bar next to \(2x\).

### Step 6: Multiply and Subtract Again
Multiply \(-12\) by the entire divisor \((x + 2)\) and subtract:
\[
(-12)(x + 2) = -12x - 24
\]
Subtract this from the result of the previous subtraction:
\[
(-12x + 5) - (-12x - 24) = 5 + 24 = 29
\]

### Step 7: Write the Final Answer
The quotient is \(2x - 12\), and the remainder is \(29\). Therefore, the final answer is:
\[
\frac{2x^2 - 8x + 5}{x + 2} = 2x - 12 + \frac{29}{x + 2}
\]

This is the result of dividing the given polynomial using long division.

Would you like any more details or have any questions?

Here are 5 related questions you might find helpful:
1. How do you perform synthetic division on the same polynomial?
2. What is the difference between polynomial long division and synthetic division?
3. How do you find the remainder when dividing polynomials?
4. Can you divide a polynomial by a polynomial of a higher degree?
5. How does the remainder theorem relate to polynomial division?

**Tip:** Always check your work by multiplying the quotient by the divisor and adding the remainder; it should give you the original dividend.

The query submitted by the user: (3x²-x²-8x+5) ÷ (x + 2) divide using polynomial long division

Learn how to divide the polynomial (3x²-x²-8x+5) by (x + 2) using polynomial long division. This detailed step-by-step solution covers key algebraic concepts like polynomial division and the remainder theorem. Suitable for students in Grades 10-12.

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