Question

Please show the synthetic division that led to those factors. For example #1a:

Answer 1

Given the function:

`f(x)=3x^4-18x^3-21x^2+144x-108`

You can find the x-intercepts by following these steps:

1. Make the function equal to zero:

`3x^4-18x^3-21x^2+144x-108=0`

2. Factor the equation:

- Notice that all the terms have 3 in common. Therefore, you can factor it out:

`3(x^4-6x^3-7x^2+48x-36)=0`

- Rewrite these terms as the Difference between two terms:

`\begin{gathered} -6x^3=-x^3-5x^3 \\ \\ -7x^2=5x^2-12x^2 \\ \\ 48x=12x+36x \end{gathered}`

Then:

`3(x^4-x^3-5x^3+5x^2-12x^2+12x+36x-36)=0`

- Make groups of two terms using parentheses:

`3\lbrack(x^4-x^3)-(5x^3-5x^2)-(12x^2-12x)+(36x-36)\rbrack=0`

- Factor the Greatest Common Factor of each group out (the greatest factor the terms have in common):

`3\lbrack x^3(x-1)-5x^2(x-1)-12x(x-1)+36(x-1)\rbrack=0`

- Factor this common factor out:

`x-1`

Then:

`3(x-1)(x^3-5x^2-12x+36)=0`

- Rewrite these terms as follows:

`\begin{gathered} -5x^2=-2x-3x^2 \\ \\ -12x=6x-18x \end{gathered}`

Then:

`3(x-1)(x^3-2x^2-3x^2+6x-18x+36)=0`

- Make groups of terms:

`3(x-1)\lbrack(x^3-2x^2)-(3x^2-6x)-(18x-36)\rbrack=0`

- Factor the Greatest Common Factor of each group out:

`3(x-1)\lbrack x^2(x-2)-3x(x-2)-18(x-2)\rbrack=0`

- Factor this factor out:

`x-2`

Then:

`3(x-1)(x-2)(x^2-3x-18)=0`

- Factor the Quadratic Expression by following two terms whose Sum is -3 and whose Product is -18. These are 3 and -6. Then:

`3(x-1)(x-2)(x+3)(x-6)=0`

3. Divide both sides of the equation by 3:

`(3(x-1)(x-2)(x+3)(x-6))/(3)=(0)/(3)`

`(x-1)(x-2)(x+3)(x-6)=0`

4. Solving for "x", you get the following values:

`\begin{gathered} x-1=0\Rightarrow x=1 \\ \\ x-2=0\Rightarrow x=2 \\ \\ x+3=0\Rightarrow x=-3 \\ \\ x-6=0\Rightarrow x=6 \end{gathered}`

Hence, knowing that you have to reject the negative values, the answer is:

`\begin{gathered} x_1=1 \\ x_2=2 \\ x_3=6 \end{gathered}`