How do I find a polynomial function f(x) of degree 3 with real coefficients?

Question

How do I find a polynomial function f(x) of degree 3 with real coefficients that has a zero of 0 and zero of 1 having multiplicity 2?

Puzzling · Accepted Answer

Write down the zeroes you want to have:
x = 0
x = 1
x = 1
(We write down x = 1 twice because it has multiplicity 2):

Rewrite those as equations equal to zero:
x = 0
x - 1 = 0
x - 1 = 0

Now simply multiply those 3 expressions together:
x(x - 1)(x - 1) = 0

That's your function:
f(x) = x(x - 1)(x - 1)
or
f(x) = x(x - 1)²

But they may want you to expand that by multiplying. First multiply (x - 1)(x - 1):
f(x) = x(x² - 2x + 1)

Then distribute the x through the parentheses:
f(x) = x^3 - 2x² + x

To verify the answer, look at the graph below. It crosses the x-axis at x=0 and touches the x-axis at x=1 (multiplicity 2).

? · Answer

x^2(x - 1) will have two zero at 0 and one zero at 1

匿名 · Answer

Use the zeroes to create three factors, and then multiply them together. Each root r gives you a factor (x-r).