Chapter 2.2, Problem 66E

To determine

To find: The polynomial function which has given zeros.

Answer to Problem 66E

The polynomial function whose zeros are -2 and 5 is $f\left(x\right)=x^2-3x-10$

Explanation of Solution

Given:

The given zeros are -2 and 5.

Concept used:

If $x={\alpha }_1,x={\alpha }_2,x={\alpha }_3,\dots x={\alpha }_n,$ are the zeros of a polynomial function $f\left(x\right)$ (say) then $\left(x-{\alpha }_1\right)$ , $\left(x-{\alpha }_2\right)$ , $,\left(x-{\alpha }_3\right),\dots ,\left(x-{\alpha }_n\right)$ are the factors of $f\left(x\right)$ that is polynomial function of degree n given by $f\left(x\right)=\left(x-{\alpha }_1\right)\left(x-{\alpha }_2\right)\left(x-{\alpha }_3\right)\dots \left(x-{\alpha }_n\right)$ .

Calculations:

The polynomial function $f\left(x\right)$ having zeros $x={\alpha }_1,x={\alpha }_2$ is

  $f\left(x\right)=\left(x-{\alpha }_1\right)\left(x-{\alpha }_2\right)$ ........... (1)

Given zeros are -2 and 5 so substituting ${\alpha }_1=-2$ and ${\alpha }_2=5$ in equation (1), the polynomial function whose zeros are -2 and 5 is

  $f\left(x\right)=\left(x-(-2)\right)\left(x-5\right)$

  $\Rightarrow f\left(x\right)=\left(x+2\right)\left(x-5\right)$

  $\Rightarrow f\left(x\right)=x^2-5x+2x-10$

  $\Rightarrow f\left(x\right)=x^2-3x-10$

Conclusion:

The polynomial function whose zeros are -2 and 5 is $f\left(x\right)=x^2-3x-10$