Polynomial Theorems (from Kutasoftware.com)

Which of the expression's binomial is a factor of the given polynomials?

• $\left(b^4-8b^3-b^2+62b-34\right)\div\left(b-7\right)$

• $\left(p^4-8p^3+10p^2+2p+4\right)\div\left(p-2\right)$

• $\left(n^4+9n^3+14n^2+50n+9\right)\div\left(n-8\right)$

• $\left(x^5+6x^4-3x^2-22x-29\right)\div\left(x+6\right)$

 $f\left(a\right)=a^3+5a^2+10a+12$ 

 at  $a=-2$  

What is the remainder of $\left(p^5+8p^4+2p^2+19p+16\right)\div\left(p+8\right)$  

For any polynomial, how many possible # of imaginary roots CANNOT have?

• 0

• 2

• 3

• 4

Which of the following is the polynomial function of least degree with integral coefficients that has the given zeros?:

 $-1,\ 2i$ 

•  $f\left(x\right)=x^3+x^2+4x+4$  

•  $f\left(x\right)=x^2-1$  

•  $f\left(x\right)=x^3+5x^2-3x-15$  

•  $f\left(x\right)=x^4-4x^3-16x-16$  

Which of the following is the polynomial function of least degree with integral coefficients that has the given zeros?:

 $2+2\sqrt{2},\ -2i$ 

•  $f\left(x\right)=x^3+x^2+4x+4$  

•  $f\left(x\right)=x^2-1$  

•  $f\left(x\right)=x^3+5x^2-3x-15$  

•  $f\left(x\right)=x^4-4x^3-16x-16$  

Which of the following is the polynomial function of least degree with integral coefficients that has the given zeros?:

 $-5,\ \sqrt{3}$ 

•  $f\left(x\right)=x^3+x^2+4x+4$  

•  $f\left(x\right)=x^2-1$  

•  $f\left(x\right)=x^3+5x^2-3x-15$  

•  $f\left(x\right)=x^4-4x^3-16x-16$  

Which of the function has zero possible # of negative real zeros according to Descartes' Rule of Signs?

• $3x^4+20x^2-32$

• $x^6-64$

• $4x^3-12x^2-5x+1$

• $9x^6-3x^5+33x^4-11x^3+18x^2-6x$

Which of the function has zero possible # of positive real zeros according to Descartes' Rule of Signs?

•  $27x^6+26x^3-1$ 

•  $8x^6+9x^3+1$ 

•  $6x^5-4x^4-63x^3+42x^2+147x-98$ 

•  $2x^5+4x^4+9x^3+18x^2-35x-70$ 

 $f\left(x\right)=5x^3-2x^2+20x-8$  

How many possible rational zeros can the following function have?