drag the expressions to the correct locations on the image. not all expressions will be used.\n\nconsider…

drag the expressions to the correct locations on the image. not all expressions will be used.\n\nconsider this quotient.\n$(x^3 - 8x + 6) \\div (x^2 - 2x + 1)$\n\nuse long division to rewrite the quotient in an equivalent form as $q(x) + \\frac{r(x)}{b(x)}$, where $q(x)$ is the quotient, $r(x)$ is the remainder, and $b(x)$ is the divisor.\n\n$x^2 - 2x + 1$\n$x^3 - 8x + 6$\n$-11x + 12$\n$x + 2$\n$-5x + 4$\n$x - 6$

drag the expressions to the correct locations on the image. not all expressions will be used.\n\nconsider this quotient.\n$(x^3 - 8x + 6) \\div (x^2 - 2x + 1)$\n\nuse long division to rewrite the quotient in an equivalent form as $q(x) + \\frac{r(x)}{b(x)}$, where $q(x)$ is the quotient, $r(x)$ is the remainder, and $b(x)$ is the divisor.\n\n$x^2 - 2x + 1$\n$x^3 - 8x + 6$\n$-11x + 12$\n$x + 2$\n$-5x + 4$\n$x - 6$

Answer

Explanation:

Step1: Divide leading terms

Divide the leading term of the dividend $x^3$ by the leading term of the divisor $x^2$. $$x^3 \div x^2 = x$$

Step2: Multiply and subtract

Multiply $x$ by $(x^2 - 2x + 1)$ and subtract from $(x^3 + 0x^2 - 8x + 6)$. $$(x^3 - 8x + 6) - (x^3 - 2x^2 + x) = 2x^2 - 9x + 6$$

Step3: Divide leading terms again

Divide the new leading term $2x^2$ by $x^2$. $$2x^2 \div x^2 = 2$$

Step4: Multiply and subtract again

Multiply $2$ by $(x^2 - 2x + 1)$ and subtract from $(2x^2 - 9x + 6)$. $$(2x^2 - 9x + 6) - (2x^2 - 4x + 2) = -5x + 4$$

Step5: Identify components

The quotient $q(x)$ is $x + 2$, the remainder $r(x)$ is $-5x + 4$, and the divisor $b(x)$ is $x^2 - 2x + 1$. $$q(x) + \frac{r(x)}{b(x)} = (x + 2) + \frac{-5x + 4}{x^2 - 2x + 1}$$

Answer:

The expressions to be dragged into the boxes are: Top box (quotient): $x + 2$ Numerator box (remainder): $-5x + 4$ Denominator box (divisor): $x^2 - 2x + 1$