Question

Write a polynomial f (x) that satisfies the given conditions. Polynomial of lowest degree with zeros of -3 (multiplicity 3) 2(multiplicity 1)and with f(0)=270 .

Answer 1

To write a polynomial f(x) with specified zeros and a value for f(0), we first express the polynomial in its factored form, then solve for the coefficient using the given f(0) value. The resulting polynomial is f(x) = -5(x + 3)^3(x - 2).

Step-by-step explanation:

To write a polynomial f(x) of the lowest degree with zeros of -3 (multiplicity 3) and 2 (multiplicity 1), we use the fact that zero of multiplicity m means the factor corresponding to that zero will appear m times in the factored form of the polynomial. Given these zeros, the polynomial in its factored form is:

f(x) = a(x + 3)^3(x - 2)

We are also given that f(0) = 270. We can use this information to find the value of a:

f(0) = a(0 + 3)^3(0 - 2) = 270

a = 270 / [(-2)(3)^3]

a = 270 / (-54)

a = -5

Now we can write the polynomial with the determined value of a:

f(x) = -5(x + 3)^3(x - 2)

This is the polynomial of the lowest degree that meets the given conditions.

Answer 2

f(x) = -5(x+3)(x+3)(x+3)(x-2)

Step-by-step explanation:

If the polynomial has three zeros in -3 and one zero in 2, the lowest degree we need is four, as the polynomial has these four zeros, so we can use a generic form of a fourth degree polynomial:

y = a(x-x1)(x-x2)(x-x3)(x-x4)

Where x1, x2, x3 and x4 are the zeros, so we have that:

y = a(x+3)(x+3)(x+3)(x-2)

Now, to find the value of the constant 'a', we need to use the information that f(0) = 270:

270 = a*3*3*3*(-2)

-54a = 270

a = -5

So the polynomial is:

f(x) = -5(x+3)(x+3)(x+3)(x-2)