Question

How do you divide $\dfrac{{({x^4} - 8{x^2} + 16)}}{{(x + 2)}}$ using synthetic division?

Answer 1

Hint: Synthetic division is a shortcut way of dividing polynomials. It gives the same results as the polynomial long division but is much faster as it involves only the coefficients of the dividend and divisor, on which we perform basic arithmetic operations. As a result, we obtain the coefficients of the quotient and the remainder.
Steps to divide a polynomial by the binomial:
> Set up the synthetic division.
> Bring down the leading coefficient to the bottom row.
> Multiply c by the value just written on the bottom row.
> Add the column created in step 3.
> Repeat until done.
> Write out the answer.

Complete step-by-step answer:
In this question, the dividend is ${x^4} - 8{x^2} + 16$ and the divisor is $x + 2$.
 Arrange the dividend in descending order and write the missing terms with coefficient 0.
$ \Rightarrow {x^4} - 8{x^2} + 16 = {x^4} + 0{x^3} - 8{x^2} + 0x + 16$
To divide by $x + 2$, we will perform the synthetic division with $x = - 2$

	${x^4}$	${x^3}$	${x^2}$	${x^1}$	${x^0}$	Row 0
	1	+0	-8	+0	+16	Row 1
+	0	-2	+4	+8	-16	Row 2
$ \times \left( { - 2} \right)$	1	-2	-4	+8	0	Row 3
	${x^3}$	${x^2}$	${x^1}$	${x^0}$	Remainder	Row 4

The values for each column for row 4 are the sum of the values in rows 2 and 3 for that column.
The values for each column of row 3 are the product of (-2) and the value in row 4 of the previous column.

Hence, the answer is ${x^3} - 2{x^2} - 4x + 8$.

Note:
Keep in mind that the division algorithm is:
Dividend=divisor (quotient) + remainder
$ \Rightarrow RHS = divisor\left( {quotient} \right) + remainder$
Put the values in the above equation.
$ \Rightarrow RHS = \left( {x + 2} \right)\left( {{x^3} - 2{x^2} - 4x + 8} \right) + 0$
Multiply it.
$ \Rightarrow RHS = {x^4} - 2{x^3} - 4{x^2} + 8x + 2{x^3} - 4{x^2} - 8x + 16$
Simplify it by applying addition and subtraction.
$ \Rightarrow RHS = {x^4} - 8{x^2} + 16$
Therefore,
$ \Rightarrow RHS = LHS$
Hence, our answer is right.