Find a 4 degree polynomial having zeros -6,-3,2 and 6 and the coefficient of x^4 equal 1
The "coefficient of x^4 equal 1" is really throwing me off.
Thanks.
To find a 4-degree polynomial with specific zeros and a coefficient of x^4 equal to 1, you can use the fact that a polynomial with zeros a, b, c, and d can be represented as:
P(x) = (x-a)(x-b)(x-c)(x-d)
In this case, the zeros given are -6, -3, 2, and 6. So the polynomial can be written as:
P(x) = (x-(-6))(x-(-3))(x-2)(x-6)
P(x) = (x+6)(x+3)(x-2)(x-6)
Now, we need to make the coefficient of x^4 equal to 1. Currently, it is not 1, since each factor contributes an additional x term. To make it 1, we need to divide the entire polynomial by a constant. In this case, we divide by the coefficient of the term with the highest degree, which is x^4.
Dividing P(x) by x^4, we get:
P(x) = (x+6)(x+3)(x-2)(x-6) / x^4
Simplifying this expression, we get:
P(x) = (x+6)(x+3)(x-2)(x-6) / (x^2)(x^2)
P(x) = (x+6)(x+3)(x-2)(x-6) / (x^2)(x^2)
So, a 4-degree polynomial with zeros -6, -3, 2, and 6 and a coefficient of x^4 equal to 1 can be written as:
P(x) = (x+6)(x+3)(x-2)(x-6) / (x^2)(x^2)
To find a 4-degree polynomial with the given zeros and the coefficient of x^4 equal to 1, we can use the fact that a polynomial in standard form can be written as:
P(x) = (x - z1)(x - z2)(x - z3)(x - z4)
where z1, z2, z3, and z4 are the zeros.
In this case, the zeros are -6, -3, 2, and 6. So, substituting these values, the polynomial can be written as:
P(x) = (x - (-6))(x - (-3))(x - 2)(x - 6)
Simplifying further:
P(x) = (x + 6)(x + 3)(x - 2)(x - 6)
Now, we can expand this expression:
P(x) = (x + 6)(x + 3)(x - 2)(x - 6)
= (x^2 + 9x + 18)(x^2 - 8x + 12)
= (x^4 - 8x^3 + 12x^2 + 9x^3 - 72x^2 + 108x + 18x^2 - 144x + 216)
= x^4 + x^3 - 45x^2 - 36x + 216
Therefore, the 4-degree polynomial with zeros -6, -3, 2, and 6 and the coefficient of x^4 equal to 1 is:
P(x) = x^4 + x^3 - 45x^2 - 36x + 216
a x ^ 4 + b x ^ 3 + c x ^ 2 + d x + e =
a ( x - x1 ) * ( x - x2 ) * ( x - x3 ) * ( x - x4 )
In your case:
a = 1
So:
x ^ 4 + b x ^ 3 + c x ^ 2 + d x + e =
( x - ( - 6 ) ) * ( x - ( - 3 ) ) * ( x - 2 ) * ( x - 6 ) =
( x + 6 ) * ( x + 3 ) * ( x - 2 ) * ( x - 6 ) =
x ^ 4 + x ^ 3 - 42 x ^ 2 -36 x + 216