Question

(lesson 6.1) (0.5 point)

1. a rectangle has a length of x and a width of 5x³ - x + 4. what is the polynomial that models the perimeter of the rectangle?

□ a. 10x³ + x + 4
□ b. 5x³ + x + 4
□ c. 10x³ + 8

(lesson 6.2) simplify the polynomial given the operation. (0.5 points each)

1. 3x²(-4x³ + 10x² - 7x + 2)
2. (2x² + x - 1)(x² + 3x)

Explanation:

Question 10

Step1: Recall perimeter formula

The perimeter $P$ of a rectangle is $P = 2(l + w)$, where $l$ is length and $w$ is width. Here $l=x$ and $w = 5x^{3}-x + 4$.

Step2: Substitute values into formula

$P=2(x+(5x^{3}-x + 4))$.

Step3: Simplify the expression inside parentheses

$x+(5x^{3}-x + 4)=5x^{3}+4$.

Step4: Multiply by 2

$P = 2(5x^{3}+4)=10x^{3}+8$.

Step1: Use distributive property

$3x^{2}(-4x^{3}+10x^{2}-7x + 2)=3x^{2}\times(-4x^{3})+3x^{2}\times10x^{2}-3x^{2}\times7x+3x^{2}\times2$.

Step2: Apply exponent - rule $a^{m}\times a^{n}=a^{m + n}$

$3x^{2}\times(-4x^{3})=-12x^{2 + 3}=-12x^{5}$, $3x^{2}\times10x^{2}=30x^{2+2}=30x^{4}$, $3x^{2}\times7x = 21x^{2+1}=21x^{3}$, $3x^{2}\times2 = 6x^{2}$.

Step3: Combine terms

The result is $-12x^{5}+30x^{4}-21x^{3}+6x^{2}$.

Step1: Use distributive property (FOIL - like for polynomials)

$(2x^{2}+x - 1)(x^{2}+3x)=2x^{2}(x^{2}+3x)+x(x^{2}+3x)-1(x^{2}+3x)$.

Step2: Distribute each term

$2x^{2}(x^{2}+3x)=2x^{2}\times x^{2}+2x^{2}\times3x=2x^{4}+6x^{3}$, $x(x^{2}+3x)=x^{3}+3x^{2}$, $-1(x^{2}+3x)=-x^{2}-3x$.

Step3: Combine like - terms

$2x^{4}+6x^{3}+x^{3}+3x^{2}-x^{2}-3x=2x^{4}+(6x^{3}+x^{3})+(3x^{2}-x^{2})-3x=2x^{4}+7x^{3}+2x^{2}-3x$.

Answer:

C. $10x^{3}+8$

Question 11