Question

Use synthetic division to show that x is a solution of the third degree polynomial, and use the result to factor de polynomial completely. List all the real solutions of the equation .

Answer 1

*Take the constant term of the divisor with the opposite sign and write it to the left.

*Write the coefficients of the dividend to the right.

Therefore, the solution is given by:

`\begin{gathered} (2x^3-13x^2+22x-8)/(x-(1)/(2))=2x^2-12x+16+(0)/(x-(1)/(2)) \\ (2x^3-13x^2+22x-8)/(x-(1)/(2))=2x^2-12x+16 \end{gathered}`

Therefore, the other solutions can be found as follows:

`2x^2-12x+16=0`

Divide both sides by 2:

`x^2-6x+8=0`

The factors of 8 that sum to -6 are -2 and -4, So:

`\begin{gathered} (x-2)(x-4)=0 \\ Hence\colon \\ x=2 \\ and \\ x=4 \end{gathered}`