Find a polynomial whose zeroes are double the zeroes of P(x)=x^3-7x^2-5x+2.
Properties of the roots of a cubic
If the roots of a cubic are a, b , and c
then start with (x-a)(x-b)(x-c)=0 and multiply out.
We get x3-(a+b+c)x2+(ab+bc+ca)x-(abc)=0
sum zeros is [-(coeffiecient of x^2 term) ]
sum of products of two at a time = coeffi
so for your equation we get
a+b+c = 7
ab + bc + ac = -5
abc = -2
let the zeros of our new equation be
2a, 2b and 2c
2a+2b+2c =
2(a+b+c) = 2(7) = 14
4ab + 4bc + 4ac = 4(ab + bc + ac) = 4(-5) = -20
(2a)(2b)(2c)
= 8abc = 8(-2) = -16
so without finding the actual roots we have
f(x) = x^3-14x^2 - 20x - 16
my second paragraph cut a bit mangled, has no effect on the rest.
should have been:
sum zeros is [-(coeffiecient of x^2 term) ]
sum of products of two at a time = coefficient of x term
product of all three roots = -constant term
so for your equation we get
etc (rest is good)
To find a polynomial whose zeroes are double the zeroes of the given polynomial P(x), we can start by finding the zeroes of P(x).
Given polynomial: P(x) = x^3 - 7x^2 - 5x + 2
To find the zeroes of P(x), we set P(x) equal to zero and solve for x:
x^3 - 7x^2 - 5x + 2 = 0
Unfortunately, the cubic equation does not have rational roots, so we need to use numerical methods to approximate the roots. One common method is to use the Newton-Raphson method or a graphing calculator.
Using numerical methods, we find the approximate roots of P(x) as follows:
x ≈ -0.62
x ≈ 1.39
x ≈ 6.23
To find a polynomial with zeroes that are double the zeroes of P(x), we take each of the roots and multiply them by 2:
2 * (-0.62) = -1.24
2 * (1.39) = 2.78
2 * (6.23) = 12.46
Now, we have the new zeroes for our desired polynomial.
Next, we construct the new polynomial by using these new zeroes. Since the roots are -1.24, 2.78, and 12.46, the polynomial is of the form:
Q(x) = (x + 1.24)(x - 2.78)(x - 12.46)
Expanding this product using FOIL or the distributive property, we get:
Q(x) = (x + 1.24)(x - 2.78)(x - 12.46)
= (x^2 - 1.54x - 3.4472)(x - 12.46)
= x^3 - 14.94x^2 + 35.2133x + 42.9001
Therefore, the polynomial Q(x) = x^3 - 14.94x^2 + 35.2133x + 42.9001 has zeroes that are double the zeroes of P(x) = x^3 - 7x^2 - 5x + 2.