# Find a polynomial of degree 3 that has zeros 1,-2, and 3 and with the coefficient of x^2 is 5.

justaguide | College Teacher | (Level 2) Distinguished Educator

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The polynomial of degree 3 has zeros 1, -2 and 3. The polynomial can be written as (x - 1)(x + 2)(x - 3)

=> (x^2 - x + 2x - 2)(x - 3)

=> (x^2 + x - 2)(x - 3)

=> x^3 + x^2 - 2x - 3x^2 - 3x + 6

=> x^3 - 2x^2 - 5x + 6

The coefficient of x^2 is 5,

=> (-5/2)*x^3 + 5x^2 + (25/2)*x - 15

The required polynomial is the polynomial is (-5/2)*x^3 + 5x^2 + (25/2)*x - 15

embizze | High School Teacher | (Level 1) Educator Emeritus

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If k is a zero of a function, then (x-k) is a factor, so

If 1 is a zero (x-1) is a factor

If -2 is a zero then (x+2) is a factor

If 3 is a zero then (x-3) is a factor.

Then a third degree polynomial with these zeros is:

f(x)=(x-1)(x+2)(x-3)

Then `f(x)=x^3-2x^2-5x+6`

In order for the coefficient on the `x^2` term to be 5, we must multiply through by `-5/2` getting

`f(x)=-5/2x^3+5x^2+25/2x-15`

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