What is a polynomial function in standard form with zeroes 1,2,-3,and -3
a x ^ 4 + b x ^ 3 + c x ^ 2 + d x + e =
a * ( x - x1 ) * ( x - x2 ) * ( x - x3 ) * ( x - x4 )
In tis case :
a * ( x - 1 ) * ( x - 2 ) * ( x + 3 ) * ( x + 3 ) =
a * ( x ^ 4 + 3 x ^ 3 - 7 x ^ 2 - 15 x + 18 )
Thank you
To find a polynomial function with given zeroes, we start by setting up the factors of the polynomial using the zeroes provided.
The zeroes are:
1, 2, -3, and -3.
These zeroes can be rewritten as (x - 1), (x - 2), (x + 3), and (x + 3), respectively.
To find the polynomial function, we multiply all the factors together:
(x - 1)(x - 2)(x + 3)(x + 3)
Expanding this expression, we get:
(x^2 - 3x + 2)(x^2 + 6x + 9)
Now, multiply the binomials using the foil method:
(x^2)(x^2) + (x^2)(6x) + (x^2)(9) + (-3x)(x^2) + (-3x)(6x) + (-3x)(9) + (2)(x^2) + (2)(6x) + (2)(9)
Simplifying further:
x^4 + 6x^3 + 9x^2 - 3x^3 - 18x^2 - 27x + 2x^2 + 12x + 18
Finally, combine like terms:
x^4 + (6x^3 - 3x^3) + (9x^2 - 18x^2 + 2x^2) + (-27x + 12x) + 18
Simplifying completely:
x^4 + 3x^3 - 7x^2 - 15x + 18
Therefore, the polynomial function, in standard form, with zeroes 1, 2, -3, and -3 is:
f(x) = x^4 + 3x^3 - 7x^2 - 15x + 18
To find a polynomial function in standard form with the given zeroes, we need to first determine the factors of the polynomial.
The zeroes of a polynomial are the values of 'x' for which the polynomial evaluates to zero. Thus, if the zeroes of the polynomial are 1, 2, -3, and -3, then the factors of the polynomial can be obtained by setting 'x' equal to each zero.
(x - 1), (x - 2), (x + 3), and (x + 3) are the factors of the polynomial, as setting each factor equal to zero gives the respective zeroes.
Now, we can multiply these factors together to get the polynomial:
P(x) = (x - 1)(x - 2)(x + 3)(x + 3)
To simplify this expression, we can combine the two identical factors (x + 3) into a single factor squared:
P(x) = (x - 1)(x - 2)(x + 3)^2
Expanding this, we get:
P(x) = (x - 1)(x - 2)(x^2 + 6x + 9)
Finally, multiplying the factors together:
P(x) = (x^3 - 3x^2 - 4x + 2)(x^2 + 6x + 9)
Expanding this expression will result in the polynomial function in standard form with zeroes 1, 2, -3, and -3.