Question

3. the graph of a function F crosses the x-axis at -1 and 3 and touches the x-axis at 5. which equations could define this function select all that apply
a- f(x)= (x+1)(x-3)(x-5)
b- f(x)= (x+1)(x-3)(x-5)^2
c- f(x)= (x+1)^4(x-3)^2(x-5)^5
d- f(x)= (x+1)^7(x-3)^3(x-5)^4

4. which polynomial functions have exactly one positive real zero select all that apply
a- f(x)= 7x^4-5x^3-4x+1
b- f(x)= 3x^4-10x^3-x-7
c- f(x)= -x^5+4x^3-2x^2-3x+7
d- f(x)= 2x^5+8x^2+4x+1

Answer 1

#3.B because of the double root
#4. Because of the one real root, the degree must be odd. So, A and B are out.

By Descartes' Rule of Signs,
A has at most 2 positive and 0 negative roots
Now apply the Rule to C and D

Answer 2

3. B and D -- f(x)=(x+1)(x-3)(x-5)^5 and f(x)=(x+1)^7(x-3)^3(x-5)^4
4. B and C -- f(x)=3x^4-10x^3-x-7 and f(x)=-x^5+4x^3-2x^2-3x+7
:)