Polynomial division by monomials practice problems from Codeyoung.

Problem: Division of Polynomials by Monomials

The task involves dividing each polynomial by a monomial. Let's solve each problem step by step.

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#### 1. $(x^2 - 2x - 11) \div (x - 5)$

This is not a division of a polynomial by a monomial but rather a division of a polynomial by another polynomial. We will use polynomial long division.

- Dividend: $x^2 - 2x - 11$
- Divisor: $x - 5$

Step 1: Divide the leading term of the dividend ($x^2$) by the leading term of the divisor ($x$):
$$
\frac{x^2}{x} = x
$$
So, the first term of the quotient is $x$.

Step 2: Multiply the entire divisor ($x - 5$) by $x$:
$$
x \cdot (x - 5) = x^2 - 5x
$$

Step 3: Subtract this result from the original dividend:
$$
(x^2 - 2x - 11) - (x^2 - 5x) = (x^2 - x^2) + (-2x + 5x) - 11 = 3x - 11
$$

Step 4: Bring down the next term (if any). Here, there are no more terms to bring down, so we proceed with the new dividend $3x - 11$.

Step 5: Divide the leading term of the new dividend ($3x$) by the leading term of the divisor ($x$):
$$
\frac{3x}{x} = 3
$$
So, the next term of the quotient is $3$.

Step 6: Multiply the entire divisor ($x - 5$) by $3$:
$$
3 \cdot (x - 5) = 3x - 15
$$

Step 7: Subtract this result from the new dividend:
$$
(3x - 11) - (3x - 15) = (3x - 3x) + (-11 + 15) = 4
$$

The remainder is $4$, and the quotient is $x + 3$. Therefore, the result is:
$$
x + 3 + \frac{4}{x - 5}
$$

Final Answer for 1:
$$
\boxed{x + 3 + \frac{4}{x - 5}}
$$

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#### 2. $(16x^4 + 12x^3 + 2x^2) \div 4x^2$

Here, we divide each term of the polynomial by the monomial $4x^2$.

- Step 1: Divide $16x^4$ by $4x^2$:
$$
\frac{16x^4}{4x^2} = 4x^{4-2} = 4x^2
$$

- Step 2: Divide $12x^3$ by $4x^2$:
$$
\frac{12x^3}{4x^2} = 3x^{3-2} = 3x
$$

- Step 3: Divide $2x^2$ by $4x^2$:
$$
\frac{2x^2}{4x^2} = \frac{1}{2}x^{2-2} = \frac{1}{2}
$$

Combine all the results:
$$
4x^2 + 3x + \frac{1}{2}
$$

Final Answer for 2:
$$
\boxed{4x^2 + 3x + \frac{1}{2}}
$$

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#### 3. $(x^2 + 5x^2 + 4x) \div 10x^2$

First, simplify the polynomial in the numerator:
$$
x^2 + 5x^2 + 4x = 6x^2 + 4x
$$

Now, divide each term by $10x^2$.

- Step 1: Divide $6x^2$ by $10x^2$:
$$
\frac{6x^2}{10x^2} = \frac{6}{10}x^{2-2} = \frac{3}{5}
$$

- Step 2: Divide $4x$ by $10x^2$:
$$
\frac{4x}{10x^2} = \frac{4}{10}x^{1-2} = \frac{2}{5}x^{-1} = \frac{2}{5x}
$$

Combine all the results:
$$
\frac{3}{5} + \frac{2}{5x}
$$

Final Answer for 3:
$$
\boxed{\frac{3}{5} + \frac{2}{5x}}
$$

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#### 4. $(10x^2 + 2x^2 + 5x) \div 10x^2$

First, simplify the polynomial in the numerator:
$$
10x^2 + 2x^2 + 5x = 12x^2 + 5x
$$

Now, divide each term by $10x^2$.

- Step 1: Divide $12x^2$ by $10x^2$:
$$
\frac{12x^2}{10x^2} = \frac{12}{10}x^{2-2} = \frac{6}{5}
$$

- Step 2: Divide $5x$ by $10x^2$:
$$
\frac{5x}{10x^2} = \frac{5}{10}x^{1-2} = \frac{1}{2}x^{-1} = \frac{1}{2x}
$$

Combine all the results:
$$
\frac{6}{5} + \frac{1}{2x}
$$

Final Answer for 4:
$$
\boxed{\frac{6}{5} + \frac{1}{2x}}
$$

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#### 5. $(4x^4 + 5x^3 + 20x^2) \div 10x^2$

Divide each term of the polynomial by the monomial $10x^2$.

- Step 1: Divide $4x^4$ by $10x^2$:
$$
\frac{4x^4}{10x^2} = \frac{4}{10}x^{4-2} = \frac{2}{5}x^2
$$

- Step 2: Divide $5x^3$ by $10x^2$:
$$
\frac{5x^3}{10x^2} = \frac{5}{10}x^{3-2} = \frac{1}{2}x
$$

- Step 3: Divide $20x^2$ by $10x^2$:
$$
\frac{20x^2}{10x^2} = \frac{20}{10}x^{2-2} = 2
$$

Combine all the results:
$$
\frac{2}{5}x^2 + \frac{1}{2}x + 2
$$

Final Answer for 5:
$$
\boxed{\frac{2}{5}x^2 + \frac{1}{2}x + 2}
$$

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#### 6. $(24x^4 + 16x^3 + 2x^2) \div 8x^2$

Divide each term of the polynomial by the monomial $8x^2$.

- Step 1: Divide $24x^4$ by $8x^2$:
$$
\frac{24x^4}{8x^2} = \frac{24}{8}x^{4-2} = 3x^2
$$

- Step 2: Divide $16x^3$ by $8x^2$:
$$
\frac{16x^3}{8x^2} = \frac{16}{8}x^{3-2} = 2x
$$

- Step 3: Divide $2x^2$ by $8x^2$:
$$
\frac{2x^2}{8x^2} = \frac{2}{8}x^{2-2} = \frac{1}{4}
$$

Combine all the results:
$$
3x^2 + 2x + \frac{1}{4}
$$

Final Answer for 6:
$$
\boxed{3x^2 + 2x + \frac{1}{4}}
$$

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Final Answers:

1. $\boxed{x + 3 + \frac{4}{x - 5}}$
2. $\boxed{4x^2 + 3x + \frac{1}{2}}$
3. $\boxed{\frac{3}{5} + \frac{2}{5x}}$
4. $\boxed{\frac{6}{5} + \frac{1}{2x}}$
5. $\boxed{\frac{2}{5}x^2 + \frac{1}{2}x + 2}$
6. $\boxed{3x^2 + 2x + \frac{1}{4}}$