Find a polynomial f(x) with leading coefficient 1 and having the given degree and zeros.
Each polynomial should be expanded from factored form, simplified and written in descending order of exponents on the variable.
For example: (x+5)(x-2) should be given as the answer x^2 + 3x - 10
degree 4; zeros -3, 1, 3, 5
please help!
X = -3, X+3 = 0.
X=1, X-1 = 0.
X=3, X-3 = 0.
X=5, X-5 = 0.
(X+3)(X-1)(X-3)(X-5)= 0,
(X^2+2X-3)(X^2-8X+15) = 0,
X^4-8X^3+15X^2+2X^3-16X^2+30X-3X^2+24X-45,
Combine like-terms and get:
X^4-6X^3-4X^2+54X-45 = 0.
To find a polynomial with a given degree and zeros, you need to start by considering each zero and creating a binomial factor in the form (x - zero).
In this case, since the zeros are -3, 1, 3, and 5, you will have four binomial factors: (x + 3), (x - 1), (x - 3), and (x - 5).
Next, multiply these binomial factors together to obtain the polynomial.
Begin by multiplying the first two factors:
(x + 3)(x - 1) = x^2 - x + 3x - 3 = x^2 + 2x - 3
Now, multiply the result by the third factor:
(x^2 + 2x - 3)(x - 3) = x^3 - 3x^2 + 2x^2 - 6x - 3x + 9 = x^3 - x^2 - 9x + 9
Lastly, multiply the result by the fourth factor:
(x^3 - x^2 - 9x + 9)(x - 5) = x^4 - 5x^3 - x^2 + 5x^2 - 9x + 45 = x^4 - 5x^3 + 4x^2 - 9x + 45
Therefore, the polynomial with degree 4 and zeros -3, 1, 3, and 5 is
f(x) = x^4 - 5x^3 + 4x^2 - 9x + 45.