Question

find each quotient using long and synthetic division. compare answers to check.
long division synthetic division

1. $(m^3 - 12m^2 + 33m)\div(m - 5)$
2. $(2x^3 - 14x + 10)\div(x + 3)$
3. $(x^4 - 2x^3 - 29x^2 - 43x + 8)\div(x - 7)$

Explanation:

For Problem 7: $(m^3 - 12m^2 + 33m) \div (m - 5)$

Long Division

Step1: Divide leading terms

$\frac{m^3}{m} = m^2$

Step2: Multiply divisor by $m^2$

$m^2(m-5) = m^3 -5m^2$

Step3: Subtract from dividend

$(m^3 -12m^2 +33m) - (m^3 -5m^2) = -7m^2 +33m$

Step4: Divide new leading term

$\frac{-7m^2}{m} = -7m$

Step5: Multiply divisor by $-7m$

$-7m(m-5) = -7m^2 +35m$

Step6: Subtract from current polynomial

$(-7m^2 +33m) - (-7m^2 +35m) = -2m$

Step7: Divide new leading term

$\frac{-2m}{m} = -2$

Step8: Multiply divisor by $-2$

$-2(m-5) = -2m +10$

Step9: Subtract to get remainder

$(-2m) - (-2m +10) = -10$

Synthetic Division

Step1: List coefficients & root

Coefficients: $1, -12, 33, 0$; Root: $5$

Step2: Bring down leading coefficient

$1$

Step3: Multiply by root, add next term

$1 \times 5 = 5; -12 +5 = -7$

Step4: Multiply by root, add next term

$-7 \times5 = -35; 33 + (-35) = -2$

Step5: Multiply by root, add last term

$-2 \times5 = -10; 0 + (-10) = -10$

Long Division

Step1: Divide leading terms

$\frac{2x^3}{x} = 2x^2$

Step2: Multiply divisor by $2x^2$

$2x^2(x+3) = 2x^3 +6x^2$

Step3: Subtract from dividend

$(2x^3 +0x^2 -14x +10) - (2x^3 +6x^2) = -6x^2 -14x$

Step4: Divide new leading term

$\frac{-6x^2}{x} = -6x$

Step5: Multiply divisor by $-6x$

$-6x(x+3) = -6x^2 -18x$

Step6: Subtract from current polynomial

$(-6x^2 -14x) - (-6x^2 -18x) = 4x$

Step7: Divide new leading term

$\frac{4x}{x} = 4$

Step8: Multiply divisor by $4$

$4(x+3) = 4x +12$

Step9: Subtract to get remainder

$(4x +10) - (4x +12) = -2$

Synthetic Division

Step1: List coefficients & root

Coefficients: $2, 0, -14, 10$; Root: $-3$

Step2: Bring down leading coefficient

$2$

Step3: Multiply by root, add next term

$2 \times (-3) = -6; 0 + (-6) = -6$

Step4: Multiply by root, add next term

$-6 \times (-3) = 18; -14 +18 = 4$

Step5: Multiply by root, add last term

$4 \times (-3) = -12; 10 + (-12) = -2$

Long Division

Step1: Divide leading terms

$\frac{x^4}{x} = x^3$

Step2: Multiply divisor by $x^3$

$x^3(x-7) = x^4 -7x^3$

Step3: Subtract from dividend

$(x^4 -2x^3 -29x^2 -43x +8) - (x^4 -7x^3) = 5x^3 -29x^2$

Step4: Divide new leading term

$\frac{5x^3}{x} = 5x^2$

Step5: Multiply divisor by $5x^2$

$5x^2(x-7) = 5x^3 -35x^2$

Step6: Subtract from current polynomial

$(5x^3 -29x^2) - (5x^3 -35x^2) = 6x^2 -43x$

Step7: Divide new leading term

$\frac{6x^2}{x} = 6x$

Step8: Multiply divisor by $6x$

$6x(x-7) = 6x^2 -42x$

Step9: Subtract from current polynomial

$(6x^2 -43x) - (6x^2 -42x) = -x +8$

Step10: Divide new leading term

$\frac{-x}{x} = -1$

Step11: Multiply divisor by $-1$

$-1(x-7) = -x +7$

Step12: Subtract to get remainder

$(-x +8) - (-x +7) = 1$

Synthetic Division

Step1: List coefficients & root

Coefficients: $1, -2, -29, -43, 8$; Root: $7$

Step2: Bring down leading coefficient

$1$

Step3: Multiply by root, add next term

$1 \times7 =7; -2 +7=5$

Step4: Multiply by root, add next term

$5 \times7=35; -29 +35=6$

Step5: Multiply by root, add next term

$6 \times7=42; -43 +42=-1$

Step6: Multiply by root, add last term

$-1 \times7=-7; 8 + (-7)=1$

Answer:

Quotient: $m^2 -7m -2$, Remainder: $-10$
$\boldsymbol{m^2 -7m -2 - \frac{10}{m-5}}$

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For Problem 8: $(2x^3 -14x +10) \div (x + 3)$