Question

drag the expressions to the correct locations on the image. not all expressions will be used. consider this quotient.
$(x^3 - 8x + 6) \div (x^2 - 2x + 1)$
use 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.
$x^3 - 8x + 6$ $-11x + 12$ $x^2 - 2x + 1$ $x - 6$ $x + 2$ $-5x + 4$

Explanation:

Step1: Divide leading terms

Divide the leading term of the dividend \(x^3 - 8x + 6\) (which is \(x^3\)) by the leading term of the divisor \(x^2 - 2x + 1\) (which is \(x^2\)). So, \(\frac{x^3}{x^2}=x\). This is the first term of the quotient \(q(x)\).

Step2: Multiply divisor by \(x\)

Multiply the divisor \(x^2 - 2x + 1\) by \(x\) to get \(x^3 - 2x^2 + x\).

Step3: Subtract from dividend

Subtract \(x^3 - 2x^2 + x\) from \(x^3 - 8x + 6\). \((x^3 - 8x + 6)-(x^3 - 2x^2 + x)=2x^2 - 9x + 6\).

Step4: Divide leading terms again

Now, divide the leading term of \(2x^2 - 9x + 6\) (which is \(2x^2\)) by the leading term of the divisor \(x^2\). So, \(\frac{2x^2}{x^2}=2\). Wait, no, wait. Wait, earlier step: Wait, the dividend is \(x^3 - 8x + 6\), divisor is \(x^2 - 2x + 1\). Wait, maybe I made a mistake. Let's redo the long division properly.

Wait, correct long division:

Divide \(x^3 - 8x + 6\) by \(x^2 - 2x + 1\).

First term: \(\frac{x^3}{x^2}=x\). Multiply divisor by \(x\): \(x(x^2 - 2x + 1)=x^3 - 2x^2 + x\). Subtract from dividend: \((x^3 - 8x + 6)-(x^3 - 2x^2 + x)=2x^2 - 9x + 6\). Now, divide \(2x^2 - 9x + 6\) by \(x^2 - 2x + 1\). The leading term: \(\frac{2x^2}{x^2}=2\). Multiply divisor by \(2\): \(2(x^2 - 2x + 1)=2x^2 - 4x + 2\). Subtract: \((2x^2 - 9x + 6)-(2x^2 - 4x + 2)= -5x + 4\). Wait, but that's not matching the options. Wait, maybe I messed up the dividend. Wait, the dividend is \(x^3 - 8x + 6\), divisor \(x^2 - 2x + 1\). Wait, maybe another approach. Wait, let's check the options. The quotient \(q(x)\) should be linear? Wait, no, the dividend is cubic, divisor is quadratic, so quotient is linear. Wait, maybe my first term was wrong. Wait, \(\frac{x^3}{x^2}=x\), then multiply divisor by \(x\): \(x^3 - 2x^2 + x\). Subtract from dividend: \(x^3 - 8x + 6 - x^3 + 2x^2 - x = 2x^2 - 9x + 6\). Now, divide \(2x^2 - 9x + 6\) by \(x^2 - 2x + 1\). The leading term is \(2\), multiply divisor by \(2\): \(2x^2 - 4x + 2\). Subtract: \(2x^2 - 9x + 6 - 2x^2 + 4x - 2 = -5x + 4\). But the options have \(-11x + 12\), etc. Wait, maybe the dividend is different? Wait, no, the problem says \((x^3 - 8x + 6) \div (x^2 - 2x + 1)\). Wait, maybe I made a mistake in the long division. Let's try again.

Wait, let's use polynomial long division:

Dividend: \(x^3 + 0x^2 - 8x + 6\) (writing \(x^3 - 8x + 6\) as \(x^3 + 0x^2 - 8x + 6\))

Divisor: \(x^2 - 2x + 1\)

First term: \(x^3 \div x^2 = x\). Multiply divisor by \(x\): \(x(x^2 - 2x + 1)=x^3 - 2x^2 + x\). Subtract from dividend:

\((x^3 + 0x^2 - 8x + 6) - (x^3 - 2x^2 + x) = 2x^2 - 9x + 6\)

Now, divide \(2x^2 - 9x + 6\) by \(x^2 - 2x + 1\). First term: \(2x^2 \div x^2 = 2\). Multiply divisor by \(2\): \(2(x^2 - 2x + 1)=2x^2 - 4x + 2\). Subtract:

\((2x^2 - 9x + 6) - (2x^2 - 4x + 2) = -5x + 4\)

But the options have \(-11x + 12\), \(x - 6\), \(x + 2\), etc. Wait, maybe the dividend is \(x^3 - 8x + 6\) and divisor is different? Wait, no, the divisor is \(x^2 - 2x + 1\). Wait, maybe I made a mistake in the problem. Wait, the options include \(x + 2\), \(x - 6\), etc. Let's try dividing \(x^3 - 8x + 6\) by \(x^2 - 2x + 1\) again, but maybe I missed a term. Wait, maybe the quotient is \(x + 2\)? Let's check: \((x + 2)(x^2 - 2x + 1) + r(x) = x^3 - 2x^2 + x + 2x^2 - 4x + 2 + r(x) = x^3 - 3x + 2 + r(x)\). Set equal to \(x^3 - 8x + 6\). So, \(x^3 - 3x + 2 + r(x) = x^3 - 8x + 6\). Then \(r(x) = -5x + 4\). But that's not matching the options. Wait, the options have \(-11x + 12\), maybe the divisor is different? Wait, no, the divisor is \(x^2 - 2x + 1\). Wait, maybe the dividend is \(x^3 - 8x + 6\) and the quotient is \(x - 6\)? Let's check: \((x - 6)(x^2 - 2x + 1) = x^3 - 2x^2 + x - 6x^2 + 12x - 6 = x^3 - 8x^2 + 13x - 6\). Subtract from \(x^3 - 8x + 6\): \((x^3 - 8x + 6) - (x^3 - 8x^2 + 13x - 6) = 8x^2 - 21x + 12\). No, that's not right. Wait, maybe the quotient is \(x + 2\), remainder \(-11x + 12\)? Let's check: \((x + 2)(x^2 - 2x + 1) = x^3 - 2x^2 + x + 2x^2 - 4x + 2 = x^3 - 3x + 2\). Then \(x^3 - 3x + 2 + (-11x + 12) = x^3 - 14x + 14\), not \(x^3 - 8x + 6\). Wait, maybe the quotient is \(x - 6\), remainder \(-11x + 12\)? Let's check: \((x - 6)(x^2 - 2x + 1) = x^3 - 2x^2 + x - 6x^2 + 12x - 6 = x^3 - 8x^2 + 13x - 6\). Then \(x^3 - 8x^2 + 13x - 6 + (-11x + 12) = x^3 - 8x^2 + 2x + 6\), not \(x^3 - 8x + 6\). Wait, maybe the dividend is \(x^3 - 8x + 6\), divisor \(x^2 - 2x + 1\), quotient \(x + 2\), remainder \(-5x + 4\), but that's not in the options. Wait, the options have \(-5x + 4\)? Yes, \(-5x + 4\) is an option. Wait, the quotient \(q(x)\) should be \(x + 2\)? Let's check: \((x + 2)(x^2 - 2x + 1) = x^3 - 2x^2 + x + 2x^2 - 4x + 2 = x^3 - 3x + 2\). Then \(x^3 - 3x + 2 + (-5x + 4) = x^3 - 8x + 6\), which matches the dividend. So, the quotient \(q(x)\) is \(x + 2\), the remainder \(r(x)\) is \(-5x + 4\), and the divisor \(b(x)\) is \(x^2 - 2x + 1\). Wait, but the problem says to rewrite as \(q(x) + \frac{r(x)}{b(x)}\). So, \(q(x)\) is \(x + 2\), \(r(x)\) is \(-5x + 4\), \(b(x)\) is \(x^2 - 2x + 1\). But the options: the first box is \(q(x)\), then the numerator of the fraction is \(r(x)\), and the denominator is \(b(x)\). Wait, the options given are \(x^3 - 8x + 6\), \(-11x + 12\), \(x^2 - 2x + 1\), \(x - 6\), \(x + 2\), \(-5x + 4\). So, \(q(x)\) is \(x + 2\), \(r(x)\) is \(-5x + 4\), \(b(x)\) is \(x^2 - 2x + 1\). So, the first box (before the plus sign) is \(x + 2\), the numerator of the fraction is \(-5x + 4\), and the denominator is \(x^2 - 2x + 1\). Wait, but let's confirm again. Let's do the long division properly:

Divide \(x^3 - 8x + 6\) by \(x^2 - 2x + 1\):

1. Divide the leading term \(x^3\) by \(x^2\) to get \(x\). Multiply the divisor by \(x\): \(x(x^2 - 2x + 1) = x^3 - 2x^2 + x\).
2. Subtract this from the dividend: \((x^3 - 8x + 6) - (x^3 - 2x^2 + x) = 2x^2 - 9x + 6\).
3. Now, divide the leading term \(2x^2\) by \(x^2\) to get \(2\). Multiply the divisor by \(2\): \(2(x^2 - 2x + 1) = 2x^2 - 4x + 2\).
4. Subtract this from the previous result: \((2x^2 - 9x + 6) - (2x^2 - 4x + 2) = -5x + 4\).
5. Since the degree of the remainder \(-5x + 4\) is less than the degree of the divisor \(x^2 - 2x + 1\), we stop.

Wait, but in step 1, we got \(x\), step 3, we got \(2\), so the quotient \(q(x)\) is \(x + 2\) (since \(x + 2\) is the sum of the terms from step 1 and step 3). The remainder \(r(x)\) is \(-5x + 4\), and the divisor \(b(x)\) is \(x^2 - 2x + 1\). So, the expression is \(x + 2 + \frac{-5x + 4}{x^2 - 2x + 1}\). Therefore, the first box (before the plus sign) is \(x + 2\), the numerator of the fraction is \(-5x + 4\), and the denominator is \(x^2 - 2x + 1\).

Answer:

The first box (before the plus sign) is \(x + 2\), the numerator of the fraction is \(-5x + 4\), and the denominator is \(x^2 - 2x + 1\). So, dragging: \(x + 2\) to the first box, \(-5x + 4\) to the numerator, and \(x^2 - 2x + 1\) to the denominator.