Computation with Polynomials: Division | EdBoost. - Free Printable

The task involves dividing polynomials using polynomial long division. Below, I will solve each problem step by step.

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Problem 1:

Divide \( x^2 - 13x + 40 \) by \( x - 8 \).

#### Step-by-Step Solution:
1. Set up the division:
\[
\require{enclose}
\begin{array}{r|rr}
x - 8 & x^2 - 13x + 40 \\
\end{array}
\]

2. Divide the leading term of the dividend by the leading term of the divisor:
\[
\frac{x^2}{x} = x
\]
Write \( x \) above the division bar.

3. Multiply \( x \) by the divisor \( x - 8 \):
\[
x \cdot (x - 8) = x^2 - 8x
\]
Write this under the dividend.

4. Subtract:
\[
(x^2 - 13x + 40) - (x^2 - 8x) = -5x + 40
\]

5. Repeat the process with the new polynomial \( -5x + 40 \):
\[
\frac{-5x}{x} = -5
\]
Write \( -5 \) above the division bar.

6. Multiply \( -5 \) by the divisor \( x - 8 \):
\[
-5 \cdot (x - 8) = -5x + 40
\]
Write this under the new polynomial.

7. Subtract:
\[
(-5x + 40) - (-5x + 40) = 0
\]

The quotient is \( x - 5 \) and the remainder is \( 0 \).

Answer:
\[
\boxed{x - 5}
\]

---

Problem 2:

Divide \( 12x^2 - 39x + 30 \) by \( 4x - 5 \).

#### Step-by-Step Solution:
1. Set up the division:
\[
\require{enclose}
\begin{array}{r|rrr}
4x - 5 & 12x^2 - 39x + 30 \\
\end{array}
\]

2. Divide the leading term of the dividend by the leading term of the divisor:
\[
\frac{12x^2}{4x} = 3x
\]
Write \( 3x \) above the division bar.

3. Multiply \( 3x \) by the divisor \( 4x - 5 \):
\[
3x \cdot (4x - 5) = 12x^2 - 15x
\]
Write this under the dividend.

4. Subtract:
\[
(12x^2 - 39x + 30) - (12x^2 - 15x) = -24x + 30
\]

5. Repeat the process with the new polynomial \( -24x + 30 \):
\[
\frac{-24x}{4x} = -6
\]
Write \( -6 \) above the division bar.

6. Multiply \( -6 \) by the divisor \( 4x - 5 \):
\[
-6 \cdot (4x - 5) = -24x + 30
\]
Write this under the new polynomial.

7. Subtract:
\[
(-24x + 30) - (-24x + 30) = 0
\]

The quotient is \( 3x - 6 \) and the remainder is \( 0 \).

Answer:
\[
\boxed{3x - 6}
\]

---

Problem 3:

Divide \( 14x^3 + 69x + 27 \) by \( 7x + 3 \).

#### Step-by-Step Solution:
1. Set up the division:
\[
\require{enclose}
\begin{array}{r|rrr}
7x + 3 & 14x^3 + 0x^2 + 69x + 27 \\
\end{array}
\]

2. Divide the leading term of the dividend by the leading term of the divisor:
\[
\frac{14x^3}{7x} = 2x^2
\]
Write \( 2x^2 \) above the division bar.

3. Multiply \( 2x^2 \) by the divisor \( 7x + 3 \):
\[
2x^2 \cdot (7x + 3) = 14x^3 + 6x^2
\]
Write this under the dividend.

4. Subtract:
\[
(14x^3 + 0x^2 + 69x + 27) - (14x^3 + 6x^2) = -6x^2 + 69x + 27
\]

5. Repeat the process with the new polynomial \( -6x^2 + 69x + 27 \):
\[
\frac{-6x^2}{7x} = -\frac{6}{7}x
\]
Write \( -\frac{6}{7}x \) above the division bar.

6. Multiply \( -\frac{6}{7}x \) by the divisor \( 7x + 3 \):
\[
-\frac{6}{7}x \cdot (7x + 3) = -6x^2 - \frac{18}{7}x
\]
Write this under the new polynomial.

7. Subtract:
\[
\left(-6x^2 + 69x + 27\right) - \left(-6x^2 - \frac{18}{7}x\right) = 69x + \frac{18}{7}x + 27 = \frac{507}{7}x + 27
\]

8. Repeat the process with the new polynomial \( \frac{507}{7}x + 27 \):
\[
\frac{\frac{507}{7}x}{7x} = \frac{507}{49}
\]
Write \( \frac{507}{49} \) above the division bar.

9. Multiply \( \frac{507}{49} \) by the divisor \( 7x + 3 \):
\[
\frac{507}{49} \cdot (7x + 3) = \frac{507}{7}x + \frac{1521}{49}
\]
Write this under the new polynomial.

10. Subtract:
\[
\left(\frac{507}{7}x + 27\right) - \left(\frac{507}{7}x + \frac{1521}{49}\right) = 27 - \frac{1521}{49} = \frac{1323}{49} - \frac{1521}{49} = -\frac{198}{49}
\]

The quotient is \( 2x^2 - \frac{6}{7}x + \frac{507}{49} \) and the remainder is \( -\frac{198}{49} \).

Answer:
\[
\boxed{2x^2 - \frac{6}{7}x + \frac{507}{49}}
\]

---

Problem 4:

Divide \( 6x^4 + 23x^2 + 10x + 1 \) by \( 3x + 1 \).

#### Step-by-Step Solution:
1. Set up the division:
\[
\require{enclose}
\begin{array}{r|rrrr}
3x + 1 & 6x^4 + 0x^3 + 23x^2 + 10x + 1 \\
\end{array}
\]

2. Divide the leading term of the dividend by the leading term of the divisor:
\[
\frac{6x^4}{3x} = 2x^3
\]
Write \( 2x^3 \) above the division bar.

3. Multiply \( 2x^3 \) by the divisor \( 3x + 1 \):
\[
2x^3 \cdot (3x + 1) = 6x^4 + 2x^3
\]
Write this under the dividend.

4. Subtract:
\[
(6x^4 + 0x^3 + 23x^2 + 10x + 1) - (6x^4 + 2x^3) = -2x^3 + 23x^2 + 10x + 1
\]

5. Repeat the process with the new polynomial \( -2x^3 + 23x^2 + 10x + 1 \):
\[
\frac{-2x^3}{3x} = -\frac{2}{3}x^2
\]
Write \( -\frac{2}{3}x^2 \) above the division bar.

6. Multiply \( -\frac{2}{3}x^2 \) by the divisor \( 3x + 1 \):
\[
-\frac{2}{3}x^2 \cdot (3x + 1) = -2x^3 - \frac{2}{3}x^2
\]
Write this under the new polynomial.

7. Subtract:
\[
\left(-2x^3 + 23x^2 + 10x + 1\right) - \left(-2x^3 - \frac{2}{3}x^2\right) = 23x^2 + \frac{2}{3}x^2 + 10x + 1 = \frac{71}{3}x^2 + 10x + 1
\]

8. Repeat the process with the new polynomial \( \frac{71}{3}x^2 + 10x + 1 \):
\[
\frac{\frac{71}{3}x^2}{3x} = \frac{71}{9}x
\]
Write \( \frac{71}{9}x \) above the division bar.

9. Multiply \( \frac{71}{9}x \) by the divisor \( 3x + 1 \):
\[
\frac{71}{9}x \cdot (3x + 1) = \frac{71}{3}x^2 + \frac{71}{9}x
\]
Write this under the new polynomial.

10. Subtract:
\[
\left(\frac{71}{3}x^2 + 10x + 1\right) - \left(\frac{71}{3}x^2 + \frac{71}{9}x\right) = 10x - \frac{71}{9}x + 1 = \frac{90}{9}x - \frac{71}{9}x + 1 = \frac{19}{9}x + 1
\]

11. Repeat the process with the new polynomial \( \frac{19}{9}x + 1 \):
\[
\frac{\frac{19}{9}x}{3x} = \frac{19}{27}
\]
Write \( \frac{19}{27} \) above the division bar.

12. Multiply \( \frac{19}{27} \) by the divisor \( 3x + 1 \):
\[
\frac{19}{27} \cdot (3x + 1) = \frac{19}{9}x + \frac{19}{27}
\]
Write this under the new polynomial.

13. Subtract:
\[
\left(\frac{19}{9}x + 1\right) - \left(\frac{19}{9}x + \frac{19}{27}\right) = 1 - \frac{19}{27} = \frac{27}{27} - \frac{19}{27} = \frac{8}{27}
\]

The quotient is \( 2x^3 - \frac{2}{3}x^2 + \frac{71}{9}x + \frac{19}{27} \) and the remainder is \( \frac{8}{27} \).

Answer:
\[
\boxed{2x^3 - \frac{2}{3}x^2 + \frac{71}{9}x + \frac{19}{27}}
\]

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Final Answers:

1. \(\boxed{x - 5}\)
2. \(\boxed{3x - 6}\)
3. \(\boxed{2x^2 - \frac{6}{7}x + \frac{507}{49}}\)
4. \(\boxed{2x^3 - \frac{2}{3}x^2 + \frac{71}{9}x + \frac{19}{27}}\)