Question

17.) $(10x^4 + 4x^3 + 10x + 14) \div (10x + 4)$
$10x+4 \overline{)10x^4 + 4x^3 + 10x + 14}$

19.) $(2p^3 - 4p^2 - 26p - 13) \div (p - 5)$
$p - 5 \overline{)2p^3 - 4p^2 - 26p - 13}$

Explanation:

Problem 17: \((10x^4 + 4x^3 + 10x + 14) \div (10x + 4)\)

Step 1: Divide the leading terms

Divide the leading term of the dividend \(10x^4\) by the leading term of the divisor \(10x\).
\(\frac{10x^4}{10x}=x^3\)
Multiply the divisor \(10x + 4\) by \(x^3\): \(x^3(10x + 4)=10x^4+4x^3\)
Subtract this from the dividend:
\((10x^4 + 4x^3 + 10x + 14)-(10x^4 + 4x^3)=10x + 14\)

Step 2: Divide the new leading terms

Now, divide the leading term of the new dividend \(10x\) by the leading term of the divisor \(10x\).
\(\frac{10x}{10x} = 1\)
Multiply the divisor \(10x + 4\) by \(1\): \(1\times(10x + 4)=10x+4\)
Subtract this from the new dividend:
\((10x + 14)-(10x + 4)=10\)

Step 3: Write the result

The quotient is \(x^3 + 1\) and the remainder is \(10\). So, we can write the division as:
\(\frac{10x^4 + 4x^3 + 10x + 14}{10x + 4}=x^3 + 1+\frac{10}{10x + 4}\) (we can simplify \(\frac{10}{10x + 4}=\frac{5}{5x + 2}\) if needed, but the form with quotient and remainder is also acceptable)

We use polynomial long - division.

Step 1: Divide the leading terms

Divide the leading term of the dividend \(2p^3\) by the leading term of the divisor \(p\).
\(\frac{2p^3}{p}=2p^2\)
Multiply the divisor \(p - 5\) by \(2p^2\): \(2p^2(p - 5)=2p^3-10p^2\)
Subtract this from the dividend:
\((2p^3 - 4p^2 - 26p - 13)-(2p^3-10p^2)=6p^2-26p - 13\)

Step 2: Divide the new leading terms

Divide the leading term of the new dividend \(6p^2\) by the leading term of the divisor \(p\).
\(\frac{6p^2}{p}=6p\)
Multiply the divisor \(p - 5\) by \(6p\): \(6p(p - 5)=6p^2-30p\)
Subtract this from the new dividend:
\((6p^2-26p - 13)-(6p^2-30p)=4p - 13\)

Step 3: Divide the new leading terms

Divide the leading term of the new dividend \(4p\) by the leading term of the divisor \(p\).
\(\frac{4p}{p}=4\)
Multiply the divisor \(p - 5\) by \(4\): \(4(p - 5)=4p-20\)
Subtract this from the new dividend:
\((4p - 13)-(4p - 20)=7\)

Step 4: Write the result

The quotient is \(2p^2+6p + 4\) and the remainder is \(7\). So,
\(\frac{2p^3 - 4p^2 - 26p - 13}{p - 5}=2p^2+6p + 4+\frac{7}{p - 5}\)

Answer:

\(x^3 + 1+\frac{10}{10x + 4}\) (or \(x^3 + 1+\frac{5}{5x + 2}\))

Problem 19: \((2p^3 - 4p^2 - 26p - 13) \div (p - 5)\)