Long division of polynomials interactive worksheet - Free Printable

Problem: Long Division of Polynomials

We are tasked with performing long division of polynomials for two problems and expressing the results in both quotient form and multiplication form. Let's solve each problem step by step.

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#### Problem 1:
Divide $ -6x^3 + 3x^2 - x + 11 $ by $ x + 2 $.

##### Step 1: Set up the division.
We write the dividend $ -6x^3 + 3x^2 - x + 11 $ and the divisor $ x + 2 $.

##### Step 2: Divide the leading term of the dividend by the leading term of the divisor.
The leading term of the dividend is $ -6x^3 $, and the leading term of the divisor is $ x $. So,
$$
\frac{-6x^3}{x} = -6x^2.
$$
This is the first term of the quotient.

##### Step 3: Multiply the entire divisor by this term and subtract from the dividend.
Multiply $ x + 2 $ by $ -6x^2 $:
$$
-6x^2 \cdot (x + 2) = -6x^3 - 12x^2.
$$
Subtract this from the original polynomial:
$$
(-6x^3 + 3x^2 - x + 11) - (-6x^3 - 12x^2) = 15x^2 - x + 11.
$$

##### Step 4: Repeat the process with the new polynomial.
The new polynomial is $ 15x^2 - x + 11 $. Divide the leading term $ 15x^2 $ by the leading term of the divisor $ x $:
$$
\frac{15x^2}{x} = 15x.
$$
This is the next term of the quotient.

Multiply $ x + 2 $ by $ 15x $:
$$
15x \cdot (x + 2) = 15x^2 + 30x.
$$
Subtract this from the current polynomial:
$$
(15x^2 - x + 11) - (15x^2 + 30x) = -31x + 11.
$$

##### Step 5: Repeat the process again.
The new polynomial is $ -31x + 11 $. Divide the leading term $ -31x $ by the leading term of the divisor $ x $:
$$
\frac{-31x}{x} = -31.
$$
This is the next term of the quotient.

Multiply $ x + 2 $ by $ -31 $:
$$
-31 \cdot (x + 2) = -31x - 62.
$$
Subtract this from the current polynomial:
$$
(-31x + 11) - (-31x - 62) = 73.
$$

##### Step 6: Write the quotient and remainder.
The quotient is $ -6x^2 + 15x - 31 $, and the remainder is $ 73 $. Therefore, the result can be written as:
$$
\text{Quotient Form: } -6x^2 + 15x - 31 + \frac{73}{x + 2}.
$$

##### Step 7: Express in multiplication form.
The multiplication form is:
$$
(x + 2)(-6x^2 + 15x - 31) + 73.
$$

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#### Problem 2:
Divide $ -8x^4 - 2x^3 + 3x^2 - 5x + 16 $ by $ 2x - 3 $.

##### Step 1: Set up the division.
We write the dividend $ -8x^4 - 2x^3 + 3x^2 - 5x + 16 $ and the divisor $ 2x - 3 $.

##### Step 2: Divide the leading term of the dividend by the leading term of the divisor.
The leading term of the dividend is $ -8x^4 $, and the leading term of the divisor is $ 2x $. So,
$$
\frac{-8x^4}{2x} = -4x^3.
$$
This is the first term of the quotient.

##### Step 3: Multiply the entire divisor by this term and subtract from the dividend.
Multiply $ 2x - 3 $ by $ -4x^3 $:
$$
-4x^3 \cdot (2x - 3) = -8x^4 + 12x^3.
$$
Subtract this from the original polynomial:
$$
(-8x^4 - 2x^3 + 3x^2 - 5x + 16) - (-8x^4 + 12x^3) = -14x^3 + 3x^2 - 5x + 16.
$$

##### Step 4: Repeat the process with the new polynomial.
The new polynomial is $ -14x^3 + 3x^2 - 5x + 16 $. Divide the leading term $ -14x^3 $ by the leading term of the divisor $ 2x $:
$$
\frac{-14x^3}{2x} = -7x^2.
$$
This is the next term of the quotient.

Multiply $ 2x - 3 $ by $ -7x^2 $:
$$
-7x^2 \cdot (2x - 3) = -14x^3 + 21x^2.
$$
Subtract this from the current polynomial:
$$
(-14x^3 + 3x^2 - 5x + 16) - (-14x^3 + 21x^2) = -18x^2 - 5x + 16.
$$

##### Step 5: Repeat the process again.
The new polynomial is $ -18x^2 - 5x + 16 $. Divide the leading term $ -18x^2 $ by the leading term of the divisor $ 2x $:
$$
\frac{-18x^2}{2x} = -9x.
$$
This is the next term of the quotient.

Multiply $ 2x - 3 $ by $ -9x $:
$$
-9x \cdot (2x - 3) = -18x^2 + 27x.
$$
Subtract this from the current polynomial:
$$
(-18x^2 - 5x + 16) - (-18x^2 + 27x) = -32x + 16.
$$

##### Step 6: Repeat the process one more time.
The new polynomial is $ -32x + 16 $. Divide the leading term $ -32x $ by the leading term of the divisor $ 2x $:
$$
\frac{-32x}{2x} = -16.
$$
This is the next term of the quotient.

Multiply $ 2x - 3 $ by $ -16 $:
$$
-16 \cdot (2x - 3) = -32x + 48.
$$
Subtract this from the current polynomial:
$$
(-32x + 16) - (-32x + 48) = -32.
$$

##### Step 7: Write the quotient and remainder.
The quotient is $ -4x^3 - 7x^2 - 9x - 16 $, and the remainder is $ -32 $. Therefore, the result can be written as:
$$
\text{Quotient Form: } -4x^3 - 7x^2 - 9x - 16 + \frac{-32}{2x - 3}.
$$

##### Step 8: Express in multiplication form.
The multiplication form is:
$$
(2x - 3)(-4x^3 - 7x^2 - 9x - 16) - 32.
$$

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Final Answers:

1. For $ \frac{-6x^3 + 3x^2 - x + 11}{x + 2} $:
- Quotient Form: $ -6x^2 + 15x - 31 + \frac{73}{x + 2} $
- Multiplication Form: $ (x + 2)(-6x^2 + 15x - 31) + 73 $

2. For $ \frac{-8x^4 - 2x^3 + 3x^2 - 5x + 16}{2x - 3} $:
- Quotient Form: $ -4x^3 - 7x^2 - 9x - 16 + \frac{-32}{2x - 3} $
- Multiplication Form: $ (2x - 3)(-4x^3 - 7x^2 - 9x - 16) - 32 $

Boxed Answers:

$$
\boxed{
\begin{aligned}
&\text{Problem 1:} \\
&\text{Quotient Form: } -6x^2 + 15x - 31 + \frac{73}{x + 2} \\
&\text{Multiplication Form: } (x + 2)(-6x^2 + 15x - 31) + 73 \\
&\text{Problem 2:} \\
&\text{Quotient Form: } -4x^3 - 7x^2 - 9x - 16 + \frac{-32}{2x - 3} \\
&\text{Multiplication Form: } (2x - 3)(-4x^3 - 7x^2 - 9x - 16) - 32
\end{aligned}
}
$$