Question

How do you divide $\left( {{x}^{4}}+81 \right)\div \left( x+3 \right)$ ?

Answer 1

Hint: We can divide $\left( {{x}^{4}}+81 \right)$ by $\left( x+3 \right)$ by simple algebraic division. First we divide the first term of dividend by first term of divisor the first term of the quotient is equal to the first of the quotient of the total division. Then we will multiply with all the terms of the divisor . Then we will subtract the result from $\left( {{x}^{4}}+81 \right)$. Now the result of subtraction is our new dividend and we will continue the same process until the highest coefficient of x in reminder is less than the highest coefficient of x in divisor.

Complete step by step answer:
We have to divide $\left( {{x}^{4}}+81 \right)$ by $\left( x+3 \right)$
First the quotient when we divide ${{x}^{4}}$ by x is equal to ${{x}^{3}}$
The first term of quotient of $\left( {{x}^{4}}+81 \right)\div \left( x+3 \right)$ is equal to ${{x}^{3}}$
Now multiplying ${{x}^{3}}$ with $\left( x+3 \right)$ we get ${{x}^{4}}+3{{x}^{3}}$
Now subtracting ${{x}^{4}}+3{{x}^{3}}$ form $\left( {{x}^{4}}+81 \right)$ we get $-3{{x}^{3}}+81$
Now diving $-3{{x}^{3}}$ by x we get $-3{{x}^{2}}$ , so $-3{{x}^{2}}$ is our next term of quotient , the quotient is updated as ${{x}^{3}}-3{{x}^{2}}$
Now multiply $-3{{x}^{2}}$ with $\left( x+3 \right)$ we get $-3{{x}^{3}}-9{{x}^{2}}$
Subtracting $-3{{x}^{3}}-9{{x}^{2}}$ from $-3{{x}^{3}}+81$ we get $9{{x}^{2}}+81$
Now diving $9{{x}^{2}}$ by x we get 9x, 9x is the next term of quotient the quotient is updated as ${{x}^{3}}-3{{x}^{2}}+9x$
Multiplying 9x with $\left( x+3 \right)$ we get $9{{x}^{2}}+27x$, subtracting $9{{x}^{2}}+27x$ from $9{{x}^{2}}+81$ we get
$-27x+81$ now we can see the next term of quotient is -27 so the quotient is updated as
${{x}^{3}}-3{{x}^{2}}+9x-27x$
Multiplying -27 with $\left( x+3 \right)$ we get $-27x-81$ and subtracting $-27x-81$ from $-27x+81$ we get 162
Now we can see the coefficient of x in 162 is less than coefficient of x in $\left( x+3 \right)$
So the quotient of $\left( {{x}^{4}}+81 \right)\div \left( x+3 \right)$ is ${{x}^{3}}-3{{x}^{2}}+9x-27x$ and reminder is 16
So we can write $\left( {{x}^{4}}+81 \right)=\left( x+3 \right)\left( {{x}^{3}}-3{{x}^{2}}+9x-27x \right)+16$

Note: Always remember we should keep the division process continued until the highest coefficient of x in the remainder is less than the highest coefficient of x in the divisor. If we terminate the division before that, then we can still divide the remainder by divisor , because the highest coefficient of x in reminder is still greater than equal to highest coefficient of x in divisor.