Question

How do you use the synthetic division to find the x intercepts of \[g(x)={{x}^{3}}-3x+2\]?

Answer 1

Hint: We are given a polynomial function and we have to find x intercepts. This means we have to find the values of x such that \[g(x)=0\]. So, we will first find a value of x so that \[g(x)=0\]. Then, we will use that factor as a monomial and then divide the given expression which will give us a quadratic equation. On solving further, we will add two more factors or the values of x for which \[g(x)=0\]. Therefore, in total we will get three values of x or we can say, we will get three x intercepts for the given polynomial function.

Complete step by step solution:
According to the given question, we are given a polynomial function and using the division method we have to find the x intercepts of the given polynomial function. In other words, we have to find values of x for which \[g(x)=0\].
The expression we have is,
\[g(x)={{x}^{3}}-3x+2\]---(1)
We will use the hit and trial method to find a value of x, for which \[g(x)=0\]. We will try values of x as \[x=0,\pm 1,\pm 2\].
For \[x=0,g(x)=2\], so not our required factor.
Next, we will take \[x=1\], we get,
\[g(x)={{1}^{3}}-3(1)+2\]
\[\Rightarrow g(x)=1-3+2=-2+2=0\]
So, \[x=1\] is a factor and so we will get the given polynomial equation by \[x-1\].
Dividing the given expression, we have,
\[x-1\overset{{{x}^{2}}+x-2}{\overline{\left){\begin{align}
  & {{x}^{3}}-3x+2 \\
 & \underline{-({{x}^{3}}-{{x}^{2}})} \\
 & {{x}^{2}}-3x+2 \\
 & \underline{-({{x}^{2}}-x)} \\
 & -2x+2 \\
 & \underline{-(-2x+2)} \\
 & \_\_\_\_0\_\_\_ \\
\end{align}}\right.}}\]
That is,
\[{{x}^{3}}-3x+2=(x-1)({{x}^{2}}+x-2)\]
We will now solve the quadratic equation, we get,
\[{{x}^{2}}+x-2\]
\[\Rightarrow {{x}^{2}}+2x-x-2\]
\[\Rightarrow x(x+2)-1(x+2)\]
\[\Rightarrow (x-1)(x+2)\]
So, we get,
\[{{x}^{2}}+x-2=(x-1)(x+2)\]
So, we get the factors of the given polynomial function as,
\[{{x}^{3}}-3x+2=(x-1)(x-1)(x+2)\]
The values of \[x=1,1,-2\]
Therefore, the coordinate of the x-intercepts are \[(1,0)\] and \[(-2,0)\].

Note: The division should be done carefully and without any mistakes and that can be confirmed by checking if the remainder is coming as 0 or not. Also, since x intercepts were asked so we took \[g(x)=0\]. But if the question had asked about y intercepts, then we will put \[x=0\].