polynomial help #2

[dmaben]

I need to write a polynomial in standard form
degree= 5
zeros= -2,-1,1,2,5
f(0)= -4

I tried to factor but instead of getting -4 i get -20 and i do not know where i am going wrong?

 

[tkhunny]

Perhaps you could multiply by 1/5?

 

[mmm4444bot]

It looks like you forgot about the unknown leading coefficient A.

If the symbols r1, r2, r3, r4, and r5 represent the five given roots (i.e., function zeroes), then we can write the polynomial function symbolically in factored form as:

f(x) = A*(x - r1)*(x - r2)*(x - r3)*(x - r4)*(x - r5)

I'm guessing that you substituted the values and then multiplied out the righthand side above (using differences of squares to ease the pain, hopefully).

f(x) = A*(x^5 - 5x^4 - 5x^3 + 25x^2 + 4x - 20)

Now let x = 0, and then solve the resulting equation for A.

 

[mmm4444bot]

It's been awhile. Did you lose interest?

Letting x = 0 gives us an equation to solve for A:

-4 = A(-20)

We find that the leading coefficient A equals 1/5.

So, we substitute 1/5 for A in our factored form and multiply (just as tkhunny predicted):

f(x) = (1/5)(x^5 - 5x^4 - 5x^3 + 25x^2 + 4x - 20)

f(x) = 1/5 x^5 - x^4 - x^3 + 5x^2 + 4/5 x - 4