Question

divide. if there is a remainder, express the result in the form q
16.) (27x³ + 72x² + 39x + 14) ÷ (9x + 3)
17.) (10x
18.) (r³ − 16r² + 73r − 87) ÷ (r − 9)
19.) (2
state if the given binomial is a factor of the given pol

Explanation:

Problem 16: \((27x^3 + 72x^2 + 39x + 14) \div (9x + 3)\)

Step 1: Use Polynomial Long Division

Divide the leading term of the dividend \(27x^3\) by the leading term of the divisor \(9x\): \(\frac{27x^3}{9x} = 3x^2\). Multiply the divisor \(9x + 3\) by \(3x^2\): \(3x^2(9x + 3)=27x^3 + 9x^2\). Subtract this from the dividend: \((27x^3 + 72x^2 + 39x + 14)-(27x^3 + 9x^2)=63x^2 + 39x + 14\).

Step 2: Continue Long Division

Divide the leading term \(63x^2\) by \(9x\): \(\frac{63x^2}{9x}=7x\). Multiply the divisor by \(7x\): \(7x(9x + 3)=63x^2 + 21x\). Subtract: \((63x^2 + 39x + 14)-(63x^2 + 21x)=18x + 14\).

Step 3: Continue Long Division

Divide the leading term \(18x\) by \(9x\): \(\frac{18x}{9x}=2\). Multiply the divisor by \(2\): \(2(9x + 3)=18x + 6\). Subtract: \((18x + 14)-(18x + 6)=8\).

Step 1: Use Polynomial Long Division

Divide the leading term \(r^3\) by \(r\): \(\frac{r^3}{r}=r^2\). Multiply the divisor \(r - 9\) by \(r^2\): \(r^2(r - 9)=r^3 - 9r^2\). Subtract from the dividend: \((r^3 - 16r^2 + 73r - 87)-(r^3 - 9r^2)=-7r^2 + 73r - 87\).

Step 2: Continue Long Division

Divide \(-7r^2\) by \(r\): \(\frac{-7r^2}{r}=-7r\). Multiply the divisor by \(-7r\): \(-7r(r - 9)=-7r^2 + 63r\). Subtract: \((-7r^2 + 73r - 87)-(-7r^2 + 63r)=10r - 87\).

Step 3: Continue Long Division

Divide \(10r\) by \(r\): \(\frac{10r}{r}=10\). Multiply the divisor by \(10\): \(10(r - 9)=10r - 90\). Subtract: \((10r - 87)-(10r - 90)=3\).

Answer:

\(3x^2 + 7x + 2+\frac{8}{9x + 3}\)

Problem 18: \((r^3 - 16r^2 + 73r - 87) \div (r - 9)\)