Question

evaluate independent practice
learning goal from lesson 6.2
i can describe the similarities between the set of integers and the system of polynomials. i can multiply polynomials. i can determine whether a set or system is closed under a given operation.
how i did (circle)
i got it! im still learning.
lesson 6.2 checkpoint
once you have completed the above problems and checked your solutions, complete the lesson checkpoint below.
complete the lesson reflection above by circling your current understanding of the learning goal.
simplify by performing the following polynomial multiplications.

1. (3x - 2)(2x² + 3x - 1)
2. -2x²(5x³ + x² - 4x - 2)

a rectangle has a length of 2x - 9 and a width of x² + 3x - 4. what is the polynomial that models the area of the rectangle?
a. x² + 5x - 13
b. 2x³ + 15x² + 19x - 36
c. 2x³ - 3x² - 35x + 36

Explanation:

Step1: Multiply first polynomial

Use distributive property:
\[

$$\begin{align*} (3x - 2)(2x^{2}+3x - 1)&=3x(2x^{2}+3x - 1)-2(2x^{2}+3x - 1)\\ &=(3x\times2x^{2}+3x\times3x - 3x\times1)-(2\times2x^{2}+2\times3x - 2\times1)\\ &=(6x^{3}+9x^{2}-3x)-(4x^{2}+6x - 2)\\ &=6x^{3}+9x^{2}-3x - 4x^{2}-6x + 2\\ &=6x^{3}+(9x^{2}-4x^{2})+(-3x - 6x)+2\\ &=6x^{3}+5x^{2}-9x + 2 \end{align*}$$

\]

Step2: Multiply second polynomial

Use distributive property:
\[

$$\begin{align*} -2x^{2}(5x^{3}+x^{2}-4x - 2)&=-2x^{2}\times5x^{3}-2x^{2}\times x^{2}+2x^{2}\times4x + 2x^{2}\times2\\ &=-10x^{5}-2x^{4}+8x^{3}+4x^{2} \end{align*}$$

\]

Step3: Find area of rectangle

Area of rectangle \(A = \text{length}\times\text{width}\), so \(A=(2x - 9)(x^{2}+3x - 4)\)
\[

$$\begin{align*} &=2x(x^{2}+3x - 4)-9(x^{2}+3x - 4)\\ &=(2x\times x^{2}+2x\times3x - 2x\times4)-(9x^{2}+9\times3x - 9\times4)\\ &=(2x^{3}+6x^{2}-8x)-(9x^{2}+27x - 36)\\ &=2x^{3}+6x^{2}-8x - 9x^{2}-27x + 36\\ &=2x^{3}+(6x^{2}-9x^{2})+(-8x - 27x)+36\\ &=2x^{3}-3x^{2}-35x + 36 \end{align*}$$

\]

Answer:

1. \(6x^{3}+5x^{2}-9x + 2\)
2. \(-10x^{5}-2x^{4}+8x^{3}+4x^{2}\)
3. C. \(2x^{3}-3x^{2}-35x + 36\)