Answer:
The complete polynomial will be:
[tex]P(x)=\frac{7}{10}x(x+3)(x-4)^2[/tex]
Step-by-step explanation:
If one root has multiplicity 2 at x = 4, the factor can write as:
[tex](x-4)^{2}[/tex] (1)
Now, another root has a multiplicity 1 at x=0 and x=-3, then the factors will be:
[tex]x[/tex] (2)
[tex](x+3)[/tex] (3)
Putting (1),(2), and (3) together we can construct the polynomial.
[tex]P(x)=Cx(x+3)(x-4)^2[/tex] (4)
C is a constant value
We need to use the point (5, 28) to find the constant C. 5 is the x value and 28 is the P(x) value. Replacing these values into the (4) equation:
[tex]28=C5(5+3)(5-4)^2[/tex]
[tex]28=C5(8)(1)^2[/tex]
[tex]28=40C[/tex]
[tex]C=\frac{28}{40}[/tex]
[tex]C=\frac{7}{10}[/tex]
Finally, the complete polynomial will be:
[tex]P(x)=\frac{7}{10}x(x+3)(x-4)^4[/tex]
I hope it helps you!
