Question

use the long division method to find the result when 4x3 + x2 272 + 18 is divided by 4x - 3. If there is a remainder, express the result in the form q(x) + r(x)/b(x)

Answer 1

`x^(2)+x-6`

Explanation:

We have the following polynomial:

`4x^3+x^2-27x+18`

We must divide it by 4x - 3, using the long division method, therefore:

`4x^3+x^2-27x+18/ \left(4x-3\right)`

We solve it below:

`\begin{gathered} \text{ We divide the leading term of the dividend by the leading term of the divisor:} \\ \\ (4x^3)/(4x)=x^2 \\ \\ \text{ We multiply it by the divisor} \\ \\ x^2\cdot(4x-3)=4x^3-3x^2 \\ \\ \text{ We subtract the dividend from the obtained result: } \\ \\ 4x^3+x^2-27x+18-4x^3+3x^2=4x^2-27x+18 \\ \\ \text{ Finally it would be:} \\ \\ x^2+(4x^2-27x+18)/(4x-3) \end{gathered}`

Now, we repeat the same procedure:

`\begin{gathered} \text{ We divide the leading term of the dividend by the leading term of the divisor:} \\ \\ (4x^2)/(4x)=x \\ \\ \text{ We multiply it by the divisor} \\ \\ x\cdot(4x-3)=4x^2-3x \\ \\ \text{ We subtract the dividend from the obtained result: } \\ \\ 4x^2-27x+18-4x^2-3x=-24x+18 \\ \\ \text{ Finally it would be:} \\ \\ x^2+x+(-24+18)/(4x-3) \end{gathered}`

We do the division for the last time and we would have the following:

`\begin{gathered} \text{ We divide the leading term of the dividend by the leading term of the divisor:} \\ \\ (-24x)/(4x)=-6 \\ \\ \text{ We multiply it by the divisor} \\ \\ -6\cdot(4x-3)=-24x+18 \\ \\ \text{ We subtract the dividend from the obtained result: } \\ \\ -24x+18-24x+18=0 \\ \\ \text{ Finally it would be:} \\ \\ x^2+x-6 \end{gathered}`

So, the correct answer is:

`4x^3+x^2-27x+18/\left(4x-3\right)=x^2+x-6`