Answer:
[tex]f(x)=x^3-12x^2+37x+130[/tex]
Step-by-step explanation:
So I'm assuming when you provided n=3, that means the degree is 3? So the first thing to know, is you can express a polynomial using it's factors as: [tex]f(x)=a(x\pm b)(x\pm b)(x\pm c)[/tex] where the sign of each factor depends on the sign of the factor... the point is it can be either. Notice the a? This usually will determine the stretch/compression of the polynomial, since sometimes the factors will have a coefficient for the x, but in this case I'm assuming all the coefficients of x are 1. So the next thing that is vital to know is that imaginary solutions come in conjugates. This means that if you have a zero at: [tex]a-bi \text{ then }a+bi \text{ is also a zero}[/tex]. So this means you have the 3 zeroes at x=-2, x=7+4i, x=7-4i. These are all the zeroes of the polynomial, since the Fundamental Theorem of Algebra states that a polynomial with degree n will have n solutions, which can be complex or real.
So the factor isn't going to be written as (x-2), it's going to be (x+2) since when you plug in -2 as x, it makes x+2 equal to 0.
The same thing applies for the two imaginary factors, so you're going to have the other two factors as (x-(7+4i)) and (x-(7-4i). You can instead think of it as ((x-7)+4i) and ((x-7)-4i) and you can use the difference of squares identity: [tex](a-b)(a+b)=a^2-b^2[/tex] where in this case a=x-7 and b=4i
So this gives you the equation: [tex]((x-7)-4i)((x-7)+4i) = (x-7)^2-(4i)^2[/tex]
Which becomes: [tex](x^2-14x+49)-16(-1) = x^2-14x+49+16 = x^2-14x+65[/tex]
So this gives us one of the factors: [tex]x^2-14x+65[/tex]
Now plug this in with the other factor and we get the equation:
[tex]f(x) = a(x+2)(x^2-14x+65)[/tex]
Now plug in 1 as x and make f(x) = 156, to solve for a
[tex]156=a(1+2)(1^2-14(1)+65)\\156=a(3)(1-14+65)\\156=a(3)(52)\\156=152a\\a=1[/tex]
So in this case, a=1, so that last step wasn't necessary, although I would check each time just in case a =/= 1.
Original equation
[tex]f(x) = (x+2)(x^2-14x+65)[/tex]
Multiply
[tex]f(x)=(x^3-14x^2+65x)+(2x^2-28x+130)\\[/tex]
Combine like terms:
[tex]f(x)=x^3+(-14x^2+2x^2)+(65x-28x)+130[/tex]
add like terms:
[tex]f(x)=x^3-12x^2+37x+130[/tex]
