Question

dividing polynomials
directions: divide each polynomial. use your answer to find the color to use for coloring the corresponding picture.

1. $(x^3 - 5x^2 - 10x + 2) \div (x + 2)$
2. $(2x^3 + 4x^2 - x - 2) \div (x + 1)$
3. $(x^3 - 13x + 12) \div (x + 4)$
4. $(x^4 + 3x^3 - 7x^2 - 8) \div (x - 2)$
5. $(3x^3 - 5x^2 + 6) \div (x - 1)$
6. $(9x^3 + 12x^2 - 8x + 15) \div (x + 2)$

(other polynomial division expressions and color - related content as shown in the chart)

Explanation:

Let's solve problem 2: \((2x^3 + 4x^2 - x - 2) \div (x + 1)\) using polynomial long division.

Step 1: Divide the leading term

Divide the leading term of the dividend \(2x^3\) by the leading term of the divisor \(x\). So, \(2x^3 \div x = 2x^2\).

Step 2: Multiply and subtract

Multiply the divisor \(x + 1\) by \(2x^2\) to get \(2x^3 + 2x^2\). Subtract this from the dividend:
\[

$$\begin{align*} &(2x^3 + 4x^2 - x - 2) - (2x^3 + 2x^2)\\ =& 2x^3 + 4x^2 - x - 2 - 2x^3 - 2x^2\\ =& 2x^2 - x - 2 \end{align*}$$

\]

Step 3: Repeat the process

Divide the leading term of the new dividend \(2x^2\) by \(x\) to get \(2x\). Multiply the divisor by \(2x\) to get \(2x^2 + 2x\). Subtract:
\[

$$\begin{align*} &(2x^2 - x - 2) - (2x^2 + 2x)\\ =& 2x^2 - x - 2 - 2x^2 - 2x\\ =& -3x - 2 \end{align*}$$

\]

Step 4: Repeat again

Divide the leading term of the new dividend \(-3x\) by \(x\) to get \(-3\). Multiply the divisor by \(-3\) to get \(-3x - 3\). Subtract:
\[

$$\begin{align*} &(-3x - 2) - (-3x - 3)\\ =& -3x - 2 + 3x + 3\\ =& 1 \end{align*}$$

\]

Step 5: Write the result

The quotient is \(2x^2 + 2x - 3\) and the remainder is \(1\). So, \((2x^3 + 4x^2 - x - 2) \div (x + 1) = 2x^2 + 2x - 3 + \frac{1}{x + 1}\)

Answer:

\(2x^2 + 2x - 3 + \frac{1}{x + 1}\)