Question

If 2 and −2 are two zeros of the polynomial (x⁴ + x³ − 34x² − 4x + 120), find all the zeros of given polynomial.

Answer 1

$\begin{array}{l}\text{Let}f\left(x\right)=x^4+x^3-34x^2-4x+120\\ \text{Since 2 and}-2\text{are the zeroes of}f\left(x\right),\text{it follows that each one of}(x-2)\text{and}\\ (x+2)\text{is a factor of}f\left(x\right).\\ \text{Consequently,}(x-2)(x+2)=(x^2-4)\text{is a factor of}f\left(x\right).\\ \text{On dividing}f\left(x\right)\text{by}(x^2-4)\text{,we get:}\end{array}$



$\begin{array}{l}\\ \therefore f\left(x\right)=0\\ =>(x^2+x-30)(x^2-4)=0\\ \begin{array}{l}=>(x^2+6x-5x-30)(x-2)(x+2)\\ \begin{array}{l}=>\left[x\right(x+6)-5(x+6\left)\right](x-2)(x+2)\\ \begin{array}{l}=>(x-5)(x+6)(x-2)(x+2)=0\\ =>x=5\text{or}x=-6\mathrm{or}x=2\text{or}x=-2\\ \text{Hence, all the zeros are 2,}-2,5\mathrm{and}-6.\end{array}\\ \end{array}\end{array}\\ \end{array}$