Question

simplify.
\\(\frac{2x^2 + 3x - 14}{x - 2}\\)
\\(\bigcirc x + 2\\)
\\(\bigcirc 2x + 7\\)
\\(\bigcirc (2x + 7)(x - 2)\\)
\\(\bigcirc x - 15\\)

question 7
1 pts
simplify.
\\((x^4 - 3x^3 - 11x^2 + 3x + 10) \div (x - 5)\\)
\\(\bigcirc x^3 + 2x^2 - x - 2\\)
\\(\bigcirc x^4 + 2x^3 - x^2 - 2x\\)
\\(\bigcirc x^3 - 8x^2 + 29x - 578\\)
\\(\bigcirc \frac{x^3 - 3x^2 - 11x + x - 2}{5}\\)

Explanation:

First Simplification Problem ( $\boldsymbol{\frac{2x^2 + 3x - 14}{x - 2}}$ )

Step1: Factor the numerator

We factor the quadratic expression $2x^2 + 3x - 14$. We need two numbers that multiply to $2\times(-14)= -28$ and add up to $3$. The numbers are $7$ and $-4$. So we rewrite the middle term:
$$2x^2 + 7x - 4x - 14$$
Then factor by grouping:
$$x(2x + 7) - 2(2x + 7) = (2x + 7)(x - 2)$$

Step2: Simplify the fraction

Now we have $\frac{(2x + 7)(x - 2)}{x - 2}$. Since $x
eq2$ (to avoid division by zero), we can cancel out the common factor $(x - 2)$:
$$\frac{(2x + 7)\cancel{(x - 2)}}{\cancel{x - 2}} = 2x + 7$$

We can use polynomial long division or synthetic division. Here we use polynomial long division.

Step1: Divide the leading terms

Divide $x^4$ by $x$ to get $x^3$. Multiply $(x - 5)$ by $x^3$: $x^4 - 5x^3$. Subtract this from the dividend:
$$(x^4 - 3x^3 - 11x^2 + 3x + 10) - (x^4 - 5x^3) = 2x^3 - 11x^2 + 3x + 10$$

Step2: Repeat the process

Divide $2x^3$ by $x$ to get $2x^2$. Multiply $(x - 5)$ by $2x^2$: $2x^3 - 10x^2$. Subtract:
$$(2x^3 - 11x^2 + 3x + 10) - (2x^3 - 10x^2) = -x^2 + 3x + 10$$

Step3: Continue

Divide $-x^2$ by $x$ to get $-x$. Multiply $(x - 5)$ by $-x$: $-x^2 + 5x$. Subtract:
$$(-x^2 + 3x + 10) - (-x^2 + 5x) = -2x + 10$$

Step4: Final step

Divide $-2x$ by $x$ to get $-2$. Multiply $(x - 5)$ by $-2$: $-2x + 10$. Subtract:
$$(-2x + 10) - (-2x + 10) = 0$$
So the quotient is $x^3 + 2x^2 - x - 2$.

Answer:

$2x + 7$ (corresponding to the option "2x + 7")

Second Simplification Problem ( $\boldsymbol{(x^4 - 3x^3 - 11x^2 + 3x + 10) \div (x - 5)}$ )