Answer:
[tex]\displaystyle \frac{1}{2}(x^3-2x^2-5x+6)[/tex]
Step-by-step explanation:
Polynomials
a)
The polynomial whose graph is shown is of third degree because it has three real roots. The roots of a polynomial are the values of x that make the expression equal to zero. We can see it happens three times in the graph provided. The roots or zeros are
x=-2, x=1, x=3
b)
The factored form of a polynomial whose roots [tex]x_1, x_2, x_3[/tex] are known is
[tex]a(x-x_1)(x-x_2)(x-x_3)[/tex]
We know the value of the roots, thus the polynomial is written as
[tex]a(x+2)(x-1)(x-3)[/tex]
We need to find the value of a. We do that by replacing the value of x=0 and finding a that makes f(0)=3 (as seen in the graph). Thus
[tex]a(0+2)(0-1)(0-3)=3[/tex]
[tex]a(2)(-1)(-3)=3[/tex]
[tex]a(6)=3[/tex]
[tex]\displaystyle a=\frac{3}{6}[/tex]
[tex]\displaystyle a=\frac{1}{2}[/tex]
Thus the factored form of the polynomial is
[tex]\displaystyle \frac{1}{2}(x+2)(x-1)(x-3)[/tex]
c)
Let's multiply all the factors
[tex]\displaystyle \frac{1}{2}(x+2)(x-1)(x-3)[/tex]
[tex]\displaystyle \frac{1}{2}(x^2+2x-x-2)(x-3)[/tex]
[tex]\displaystyle \frac{1}{2}(x^2+x-2)(x-3)[/tex]
[tex]\displaystyle \frac{1}{2}(x^3-3x^2+x^2-3x-2x+6)[/tex]
[tex]\boxed{ \displaystyle \frac{1}{2}(x^3-2x^2-5x+6)}[/tex]
