Polynomial Functions

If  $f\left(x\right)=5x^2-3\ \ and\$ 
 $g\left(x\right)=-2x^3+3x-1,$  
then what is  $f\left(x\right)\times g\left(x\right)?$  

•  $-10x^5-5x^2+6x^3+3$  

•  $-10x^5+21x^3-5x^2-9x+3$  

•  $-10x^6+21x^3+5x^2+3$  

•  $-10x^6+15x^3+6x^2-3$  

Which of the following functions are polynomials? Select all that apply.

• $y=x^2-3x+2$

• $y=\frac{x^2-3x+2}{2}$

• $y=2$

• $y=5x^{-3}-2x$

• $y=5x^6-7x^3+x-8$

Let  $f\left(x\right)=6x^4-2x^3+x-3\ and\$ 
 $g\left(x\right)=-3x^4+5x^2-10x+7.$  
Then,  $f\left(x\right)+g\left(x\right)=$  

•  $3x^4-2x^3+5x^2-9x+4$  

•  $3x^4-7x^3-9x+10$  

•  $3x^4-2x^3-5x^2+11x-10$  

•  $9x^4-2x_{ }^3+5x^2-9x+4$  

Let  $f\left(x\right)=6x^4-2x^3+x-3\ and\$ 
 $g\left(x\right)=-3x^4+5x^2-10x+7.$  
Then,  $f\left(x\right)-g\left(x\right)=$  

•  $3x^4-2x^3-5x^2+11x-10$  

•  $9x^4-2x^3-5x^2+11x-10$  

•  $9x^4-2x^5+5x^2-9x+4$  

•  $3x^4-2x^3+5x^2-9x+4$  

Simplify  $\left(x-4\right)^3.$  

•  $x^3+12x^2+48x-64$  

•  $64-48x+12x^2-x^3$  

•  $x^3+12x^2-16x-4$  

•  $x^3-12x^2+48x-64$  

Simplify the expression  $x^4-81$  using your knowledge of polynomial identities.

•  $\left(x-4\right)\left(x^2-16\right)$  

•  $\left(x^2-3\right)\left(x^2+3\right)\left(x+3\right)$  

•  $\left(x^2-9\right)\left(x+9\right)$  

•  $\left(x+3\right)\left(x-3\right)\left(x^2+9\right)$  

Simplify the expression  $\left(4x^2+9\right)^2$  using your knowledge of polynomial identities.

•  $\left(4x^2-9\right)^2+144x^2$  

•  $\left(2x-3\right)^2+12x^2$  

•  $\left(4x^2-9\right)^2+5184x^2$  

•  $\left(2x^2-3\right)^2+144x^2$  

Simplify the expression  $\left(2y+5z\right)^3$  using your knowledge of polynomial identities.

•  $2y^3+30y^2z+30yz^2+5x^3$  

•  $8y^3+60y^2+125z^3$  

•  $8y^3+60y^2z+150yz^2+125z^3$  

•  $2y^3+120y^2z+30yz^2+125z^3$  

What are the solutions to the equation  $f\left(x\right)=x^3-125?$  

•  $x=5,\ -5,\ \frac{5i}{2}$  

•  $x=5,\ \frac{-5+5i\sqrt{3}}{2},\ \frac{-5-5i\sqrt{3}}{2}$  

•  $x=5,\ \frac{-5+\sqrt{75}}{2},\ \frac{-5-\sqrt{75}}{2}$  

•  $x=5,\ 25i,\ -25i$  

Find the solution(s) to the equation  $y=x^4-16.$  Select all that apply.

•  $x=-2$  

•  $x=2$  

•  $x=i$  

•  $x=2i$  

•  $x=-2i$