Question

1. select two polynomials from this list to the right and circle them!
2. multiply your two polynomials using the rectangle below! it’s a giant multiplication problem.
3. once you have filled in the boxes, color like terms the same color. for example, all of the x² terms would be the same color and all of the x terms would be a different color. the colored boxes should form a diagonal pattern.
4. combine like terms and write the simplified product of the polynomials on the line under the rectangle.

x⁵ + 3x⁴+ 8x³ - 3x²+ 4x - 2
4x⁵ - 2x⁴+ 3x³ + 2x²- 8x + 10
3x⁵ - 6x⁴ - 9x³ + x²+ 10x - 7
2x⁵ + 5x⁴+ 6x³ - 4x²- 6x - 3

Explanation:

Step1: Select two polynomials

Let's choose \(p(x)=x^{5}+3x^{4}+8x^{3}-3x^{2}+4x - 2\) and \(q(x)=2x^{5}+5x^{4}+6x^{3}-4x^{2}-6x - 3\)

Step2: Multiply each term of first polynomial by second

\((x^{5}+3x^{4}+8x^{3}-3x^{2}+4x - 2)\times(2x^{5}+5x^{4}+6x^{3}-4x^{2}-6x - 3)\)
\[

$$\begin{align*} x^{5}\times(2x^{5}+5x^{4}+6x^{3}-4x^{2}-6x - 3)&=2x^{10}+5x^{9}+6x^{8}-4x^{7}-6x^{6}-3x^{5}\\ 3x^{4}\times(2x^{5}+5x^{4}+6x^{3}-4x^{2}-6x - 3)&=6x^{9}+15x^{8}+18x^{7}-12x^{6}-18x^{5}-9x^{4}\\ 8x^{3}\times(2x^{5}+5x^{4}+6x^{3}-4x^{2}-6x - 3)&=16x^{8}+40x^{7}+48x^{6}-32x^{5}-48x^{4}-24x^{3}\\ - 3x^{2}\times(2x^{5}+5x^{4}+6x^{3}-4x^{2}-6x - 3)&=-6x^{7}-15x^{6}-18x^{5}+12x^{4}+18x^{3}+9x^{2}\\ 4x\times(2x^{5}+5x^{4}+6x^{3}-4x^{2}-6x - 3)&=8x^{6}+20x^{5}+24x^{4}-16x^{3}-24x^{2}-12x\\ -2\times(2x^{5}+5x^{4}+6x^{3}-4x^{2}-6x - 3)&=-4x^{5}-10x^{4}-12x^{3}+8x^{2}+12x + 6 \end{align*}$$

\]

Step3: Combine like - terms

\[

$$\begin{align*} \text{For }x^{10}\text{ term: }&2x^{10}\\ \text{For }x^{9}\text{ terms: }&5x^{9}+6x^{9}=11x^{9}\\ \text{For }x^{8}\text{ terms: }&6x^{8}+15x^{8}+16x^{8}=37x^{8}\\ \text{For }x^{7}\text{ terms: }&-4x^{7}+18x^{7}-6x^{7}=8x^{7}\\ \text{For }x^{6}\text{ terms: }&-6x^{6}-12x^{6}+48x^{6}+8x^{6}=38x^{6}\\ \text{For }x^{5}\text{ terms: }&-3x^{5}-18x^{5}-32x^{5}-18x^{5}+20x^{5}-4x^{5}=-55x^{5}\\ \text{For }x^{4}\text{ terms: }&-9x^{4}-48x^{4}+12x^{4}+24x^{4}-10x^{4}=-31x^{4}\\ \text{For }x^{3}\text{ terms: }&-24x^{3}+18x^{3}-16x^{3}-12x^{3}=-34x^{3}\\ \text{For }x^{2}\text{ terms: }&9x^{2}-24x^{2}+8x^{2}=-7x^{2}\\ \text{For }x\text{ terms: }&-12x + 12x=0\\ \text{Constant term: }&6 \end{align*}$$

\]

Answer:

\(2x^{10}+11x^{9}+37x^{8}+8x^{7}+38x^{6}-55x^{5}-31x^{4}-34x^{3}-7x^{2}+6\)