Question: part C and D:
Solution:
C) Find the remaining zeros of f(x):
Let the following polynomial function:
[tex]f(x)=8x^4-20x^3-14x^2+5x+3[/tex]
Remember that the roots or zeros are the y-intersections of the function, that is when y=0. Now, to find the zeros of this function we must factor the following expression:
[tex]8x^4-20x^3-14x^2+5x+3=0[/tex]
the factor of this expression is:
[tex](x-3)(2x+1)^2(2x-1)=0[/tex]
now, applying the zero factor theorem, we get the following:
[tex]x-3\text{ = 0}[/tex]
or
[tex]2x+1=0[/tex]
or
[tex]2x-1=0[/tex]
solving for each one of the above equations we get:
[tex]x\text{ = }3[/tex]
or
[tex]x\text{ = }-\frac{1}{2}[/tex]
or
[tex]x\text{ = }\frac{1}{2}[/tex]
then, the zeros (roots or solutions) are:
[tex]x\text{ }=\text{ 3, x = -}\frac{1}{2},\text{ x = }\frac{1}{2}[/tex]
D) write the complete linear factorization of f(x)
According to the above results, we get that the complete linear factorization of f(x) is:
[tex]f(x)=(x-3)(2x+1)(2x+1)^{}(2x-1)[/tex]
