Question

1. the area of a triangle is $15x^4 + 3x^3 + 4x^2 - x - 3$ square meters. the length of the base of the triangle is $6x^2 - 2$ meters. what is the height of the triangle?

problems 10 – 13: simplify each expression.

1. $(8x^4 - 4x^3 + 4x^2 - 8x + 3) \div (2x - 1)$
2. $(2x^4 - x^3 + 8x^2 - 4x + 3) \div (2x^2 - x)$
3. $\frac{2x^3 + 3x^2 + 9x + 30}{x^2 - x + 7}$
4. $\frac{5x^4 + x^3 - 10x^2 - 2x}{5x + 1}$

Explanation:

Problem 9

Step1: Recall triangle area formula

The area of a triangle is $A = \frac{1}{2}bh$, where $A$ is area, $b$ is base, $h$ is height. Rearrange to solve for $h$: $h = \frac{2A}{b}$

Step2: Substitute given expressions

Substitute $A = 15x^4 + 3x^3 + 4x^2 - x - 3$ and $b = 6x^2 - 2$:
$h = \frac{2(15x^4 + 3x^3 + 4x^2 - x - 3)}{6x^2 - 2}$

Step3: Factor numerator and denominator

Factor denominator: $6x^2 - 2 = 2(3x^2 - 1)$
Factor numerator: $2(15x^4 + 3x^3 + 4x^2 - x - 3) = 2[(15x^4 - 5x^2) + (3x^3 - x) + (9x^2 - 3)] = 2[5x^2(3x^2-1) + x(3x^2-1) + 3(3x^2-1)] = 2(3x^2-1)(5x^2 + x + 3)$

Step4: Cancel common factors

$h = \frac{2(3x^2-1)(5x^2 + x + 3)}{2(3x^2 - 1)} = 5x^2 + x + 3$

Step1: Use polynomial long division

Divide $8x^4 - 4x^3 + 4x^2 - 8x + 3$ by $2x - 1$
First term: $\frac{8x^4}{2x} = 4x^3$. Multiply $(2x-1)$ by $4x^3$: $8x^4 - 4x^3$. Subtract from dividend:
$(8x^4 - 4x^3 + 4x^2 - 8x + 3) - (8x^4 - 4x^3) = 4x^2 - 8x + 3$

Step2: Divide new polynomial

Next term: $\frac{4x^2}{2x} = 2x$. Multiply $(2x-1)$ by $2x$: $4x^2 - 2x$. Subtract:
$(4x^2 - 8x + 3) - (4x^2 - 2x) = -6x + 3$

Step3: Divide remaining term

Final term: $\frac{-6x}{2x} = -3$. Multiply $(2x-1)$ by $-3$: $-6x + 3$. Subtract:
$(-6x + 3) - (-6x + 3) = 0$

Step1: Use polynomial long division

Divide $2x^4 - x^3 + 8x^2 - 4x + 3$ by $2x^2 - x$
First term: $\frac{2x^4}{2x^2} = x^2$. Multiply $(2x^2 - x)$ by $x^2$: $2x^4 - x^3$. Subtract from dividend:
$(2x^4 - x^3 + 8x^2 - 4x + 3) - (2x^4 - x^3) = 8x^2 - 4x + 3$

Step2: Divide new polynomial

Next term: $\frac{8x^2}{2x^2} = 4$. Multiply $(2x^2 - x)$ by $4$: $8x^2 - 4x$. Subtract:
$(8x^2 - 4x + 3) - (8x^2 - 4x) = 3$

Answer:

$5x^2 + x + 3$ meters

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Problem 10