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Question

The polynomial of degree 4, P(x)P(x) has a root of multiplicity 2 at x=1x=1 and roots of multiplicity 1 at x=0x=0 and x=−3x=-3. It goes through the point (5,448)(5,448).

b=

c=

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Answer 1

take each root, change its sign, then stick an x in front of it. give each factor the exponent that's the same as the multiplicity. then form a product of all those factors. Stick an unknown coefficient "a" in front of them.

y = a(x-0)(x+3)(x-1)^2

plug in the point (5,448) for (x,y) and solve for "a"

448 = a(5)(8)(16)

a = 448/640= 0.7 = 7/10

f(x) = (7/10)x(x-1)^2(x+3)

expand to get the more explicit 4th degree polynomial, in the standard ax^4 +b^x3 +cx^2 + dx+ e form

if you want. that's probably/possibly what you mean by finding b and c, the coefficients of the cubic and squared terms. "a" the coefficient of the 4th degree term will be 7/10. "e" the constant term will be 0, the "d" term will 21/10 or 2.1. that leaves b and c the coefficients of the 3 and 2 degree terms. Multiply out the factors to find them.

f(x) = (.7)x(x^2-2x+1)(x+3) = .7x(x^3+x^2-5x+3) = .7x^4 +.7x^3 -3.5x^2+2.1x

b= .7 = coefficient of the cubic term

c=-3.5 = coefficient of the quadratic, squared term

a=.7= coefficient of the 4th degree term

d=2.1 = coefficient of the linear term

e=0= the constant term

no guarantees the above doen't have an arithmetic mistake somewhere, but this is the basic method to solve for the 4th degree polynomial.

A "nicer" way to write the equation or polynomial, without fractions, would be

10y = 7x^4 +7x^3 -35x^2 +21x