Question

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

Answer 1

Hint: To use the synthetic division method for division, we need to use the following steps:
\[1.\]Write down the coefficients of the dividend (insert dummy variable if necessary).
\[2.\]Change the sign of the constant of the divisor or write down the divisor is equal to zero, to find the value of root.
\[3.\]Bring down the first coefficient of the dividend.
\[4.\]Multiply it and then add to the above coefficient; continue the same process.
\[5.\]The answer is the sequence of coefficients of the new polynomial but one degree less than the original polynomial.
\[6.\]The last term is the divisor; we need to put it over the divisor.

Complete step-by-step solution:
Given question is as following:
\[({x^4} - 8{x^2} + 16) \div (x + 2)\]
Or, we can write it down as:
\[\dfrac{{({x^4} - 8{x^2} + 16)}}{{(x + 2)}}\].
So, dividend of the given question is:
\[({x^4} - 8{x^2} + 16)\].
So, we can write it down with the following expanded form:
\[({x^4} + 0.{x^3} - 8{x^2} + 0.x + 16)\].
So, it is very clear from the above equation that the coefficients of the \[{x^4},{x^3},{x^2},x\] and \[{x^0}\]are \[1,0, - 8,0\]and \[16\] respectively.
And, the divisor of the given question is:
\[(x + 2)\].
So, the constant term in the divisor to perform the synthetic division will be the root of \[(x + 2) = 0\].
So, the constant term will be,
\[(x + 2) = 0\]
\[ \Rightarrow x = - 2.\]
So, we need to write down the above terms as following way:
\[ - 2\left| \!{\underline {\,
  {\begin{array}{*{20}{c}}
  1&0&{ - 8}&0&{16}
\end{array}} \,}} \right. \]
Now, we will write down the first term directly as following:
 \[ - 2\left| \!{\underline {\,
  {\begin{array}{*{20}{c}}
  1&0&{ - 8}&0&{16} \\
  {\overline 1 }&{}&{}&{}&{}
\end{array}} \,}} \right. \]
Now, we will cross multiply it by \[ - 2\] and then add it to the next term; we will repeat this for next all terms:
$ - 2\left| \!{\underline {\,
  {\begin{array}{*{20}{c}}
  1&0&{ - 8}&0&{16} \\
  {\overline 1 }&{\overline { - 2} }&{\overline 4 }&{\overline 8 }&{\overline { - 16} }
\end{array}} \,}} \right. $
$ - 2\left| \!{\underline {\,
  {\begin{array}{*{20}{c}}
  1&{ - 2}&{ - 4}&8&0
\end{array}} \,}} \right. $
So, we need to write the above terms as the coefficient of terms starts with the one degree less from the highest degree (given in the question as \[4\]).
So, the term would become \[ = 1.{x^3} + ( - 2).{x^2} + ( - 4).{x^1} + 8.{x^0} + 0 = ({x^3} - 2{x^2} - 4x + 8)\] .
So, the quotient of the division will be \[({x^3} - 2{x^2} - 4x + 8)\] and the remainder will be \[0\] , as the term under the constant term after addition is \[0\] .
So, we can rewrite it as following:
\[({x^4} - 8{x^2} + 16) \div (x + 2)\]
\[ = ({x^3} - 2{x^2} - 4x + 8) + \dfrac{0}{{(x + 2)}}\]
\[ = ({x^3} - 2{x^2} - 4x + 8).\]

\[\therefore \] The quotient of the division will be \[({x^3} - 2{x^2} - 4x + 8)\] and the remainder will be \[0\] .

Note: Points to remember:
To apply the synthetic division, the degree of the \[x\] in the divisor should be \[1\] and the coefficient of \[x\]in the divisor shall be variable.
Also, we can cross check the above division by applying the following method:
Let's say, \[f(x) = ({x^4} - 8{x^2} + 16)\].
Now, we need to put
\[(x + 2) = 0.\]
Now, taking the constant term into R.H.S, we get:
\[x = - 2.\]
Now, put this value in above function, we get:
So, \[f( - 2) = ({( - 2)^4} - 8{( - 2)^2} + 16).\]
By simplifying, we get:
\[f( - 2) = (16 - 32 + 16) = 0.\]
So, it is clear that the value of \[f( - 2)\]is equal to the remainder of the final division that we have performed above.
So, at the root value of \[(x + 2) = 0\], it leaves us the same remainder of the given function.
So, we can imply that our final division is correct.