Question

simplify (6x² - 3 + 5x³) - (4x³ - 2x² - 16).
2x³ - x² + 21
-2x³ + x² - 21
x³ + 8x² + 13
-x³ - 8x² + 13
question 8(multiple choice worth 1 points) (05.03 lc)
choose the correct simplification of the expression $\frac{3b}{a^{-2}}$.
3a²b
$\frac{a^{2}}{3b}$
$\frac{3a^{2}}{b}$
already simplified
question 9(multiple choice worth 1 points) (06.02 mc)
the following table shows the length and width of a rectangle.

	length	width
rectangle a	3x + 5	2x - 3

which expression is the result of the perimeter of rectangle a and demonstrates the closure property?

Explanation:

First question:

Step1: Remove parentheses

$(6x^{2}-3 + 5x^{3})-(4x^{3}-2x^{2}-16)=6x^{2}-3 + 5x^{3}-4x^{3}+2x^{2}+16$

Step2: Combine like - terms for $x^{3}$ terms

$5x^{3}-4x^{3}=x^{3}$

Step3: Combine like - terms for $x^{2}$ terms

$6x^{2}+2x^{2}=8x^{2}$

Step4: Combine constant terms

$-3 + 16=13$
The simplified form is $x^{3}+8x^{2}+13$.

Second question:

Step1: Use the negative - exponent rule $a^{-n}=\frac{1}{a^{n}}$

$\frac{3b}{a^{-2}}=3b\times a^{2}=3a^{2}b$

Third question:

Step1: Recall the perimeter formula for a rectangle $P = 2(l + w)$

Here, $l = 3x+5$ and $w = 2x - 3$.
$P=2((3x + 5)+(2x-3))$

Step2: Simplify the expression inside the parentheses

$(3x + 5)+(2x-3)=3x+5+2x - 3=5x+2$

Step3: Multiply by 2

$P = 2(5x+2)=10x + 4$. Since the sum of two polynomials ($3x + 5$ and $2x-3$) is a polynomial ($5x + 2$) and the product of a polynomial ($5x + 2$) and a monomial (2) is a polynomial ($10x + 4$), it demonstrates the closure property of polynomials under addition and multiplication.

Answer:

First question:

C. $x^{3}+8x^{2}+13$

Second question:

A. $3a^{2}b$

Third question:

$10x + 4$