Find a polynomial with the zeros -1, 7, sqrt2, and -(sqrt2).
Can this be f(x) = x^4 - 6x^3 - 9x^2 + 12x + 14?
To find a polynomial with the given zeros, we can use the fact that if a number (let's call it "a") is a zero of a polynomial, then (x - a) is a factor of the polynomial.
Given that the zeros are -1, 7, sqrt2, and -(sqrt2), we can write the factors of the polynomial as follows:
(x - (-1))(x - 7)(x - sqrt2)(x - -(sqrt2))
The last two factors (x - sqrt2) and (x - -(sqrt2)) can be simplified to (x - sqrt2) and (x + sqrt2), respectively:
(x + 1)(x - 7)(x - sqrt2)(x + sqrt2)
Multiplying these factors together will give us the desired polynomial. Let's expand it:
(x + 1)(x - 7)(x - sqrt2)(x + sqrt2)
= (x^2 - 6x - 7)(x^2 - 2)
Expanding further:
(x^2 - 6x - 7)(x^2 - 2)
= x^4 - 6x^3 - 7x^2 - 2x^2 + 12x + 14
Now we can simplify and rearrange the terms to match the given polynomial:
x^4 - 6x^3 - 7x^2 - 2x^2 + 12x + 14
= x^4 - 6x^3 - 9x^2 + 12x + 14
Therefore, the given polynomial f(x) = x^4 - 6x^3 - 9x^2 + 12x + 14 is indeed correct.
well, it could be
(x+1)(x-7)(x^2-2)
= what you wrote.
extra credit: are there any others?