Question

Find a polynomial with zeros at x=2,x=-1 and x=0

Answer 1

f(x) = x³ - x² - 2x

Explanation:

given a polynomial with zeros x = a, x = b, x = c

Then the factors are (x - a), (x - b) and (x - c)

and the polynomial is the product of the factors, that is

f(x) = k(x - a)(x - b)(x - c) ← where k is a multiplier

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Here the zeros are x = 2, x = - 1 and x = 0, thus the factors are

(x - 2), (x + 1) and (x - 0) , thus

y = kx(x - 2)(x + 1) ← let k = 1 and expand factors

= x(x² - x - 2) = x³ - x² - 2x

Hence a possible polynomial is

f(x) = x³ - x² - 2x

Answer 2

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

Explanation:

If x = a is a zero of a polynomial, then x-a is a factor of the polynomial. Given the factors of a polynomial, the polynomial can be obtained by multiplying the factors.

The factors of the given polynomial are;

x - 2

x + 1

x

Multiplying the first two factors;

(x-2)(x+1) = x^2 + x -2x -2

= x^2 -x -2

We finally multiply this result by x to obtain our polynomial;

f(x) = x ( x^2 -x -2)

= x^3 - x^2 -2x

which is a cubic polynomial since it has 3 roots.