use the remainder theorem an the factor theorem to determine whether (y-3) is a factor of (y4+2y2-4).
evaluate the polynomial at y=3.
3^4+2*3^2-4=81+18-4=95 which is not zero.
y-3 is not a factor, or the remainder would be zero.
To determine whether (y-3) is a factor of (y^4 + 2y^2 - 4), we can use both the remainder theorem and the factor theorem.
1. Remainder Theorem:
According to the remainder theorem, if (y-3) is a factor of the polynomial, then the remainder when (y^4 + 2y^2 - 4) is divided by (y-3) should be zero.
To check this, we substitute y = 3 into the polynomial:
(3^4 + 2(3^2) - 4) = (81 + 18 - 4) = 95
Since the remainder is not zero, (y-3) is not a factor of (y^4 + 2y^2 - 4) according to the remainder theorem.
2. Factor Theorem:
According to the factor theorem, if (y-3) is a factor of the polynomial, then (y-3) should be a solution to the polynomial equation (y^4 + 2y^2 - 4) = 0.
To check this, we substitute y = 3 into the polynomial equation:
(3^4 + 2(3^2) - 4) = (81 + 18 - 4) = 95
Since the equation does not equal zero, (y-3) is not a factor of (y^4 + 2y^2 - 4) according to the factor theorem.
Therefore, based on both the remainder theorem and the factor theorem, we can conclude that (y-3) is not a factor of (y^4 + 2y^2 - 4).
To determine whether (y-3) is a factor of (y^4 + 2y^2 - 4), we can use the remainder theorem and the factor theorem.
1. Remainder Theorem:
According to the remainder theorem, if we divide a polynomial f(x) by a binomial (x - a), the remainder will be f(a).
In our case, if we divide (y^4 + 2y^2 - 4) by (y - 3), the remainder will be (y = 3).
2. Factor Theorem:
According to the factor theorem, if we substitute a value a into a polynomial function f(x) and get f(a) = 0, then (x - a) is a factor of f(x).
In our case, if we substitute y = 3 into (y^4 + 2y^2 - 4), and we find that (3^4 + 2(3^2) - 4) = 0, then (y - 3) is a factor of (y^4 + 2y^2 - 4).
Now let's calculate the value of (y^4 + 2y^2 - 4) at y = 3:
(3^4 + 2(3^2) - 4) = (81 + 18 - 4) = 95
Since the remainder is not zero, we can conclude that (y - 3) is not a factor of (y^4 + 2y^2 - 4).