Answer:
[tex]P(x)=x^3+12x^2+45x+50[/tex]
Step-by-step explanation:
Roots of a polynomial
If we know the roots of a polynomial, say x1,x2,x3,...,xn, we can construct the polynomial using the formula
[tex]P(x)=a(x-x_1)(x-x_2)(x-x_3)...(x-x_n)[/tex]
Where a is an arbitrary constant.
We are given the following roots:
-5 multiplicity 2 (-5 twice)
-2 multiplicity 1
Thus, the polynomial is:
[tex]P(x)=a(x-(-5))(x-(-5))(x-(-2))[/tex]
[tex]P(x)=a(x+5)(x+5)(x+2)[/tex]
We are not given any clue about the value of a, so we choose a=1. Multiplying:
[tex]P(x)=(x^2+5x+5x+25)(x+2)=(x^2+10x+25)(x+2)[/tex]
[tex]P(x)=x^3+2x^2+10x^2+20x+25x+50[/tex]
Simplifying:
[tex]\mathbf{P(x)=x^3+12x^2+45x+50}[/tex]
