Form a third degree polynomial function with real coefficients such that -7 + i and -9 are zeros
the -7 + i has a conjugate partner of -7 - i
so (-7+i)(-7-i)
= 49 - i^2
= 50
-7+i + (-7-i)
= -14
so we can form the quadratic as
x^2 + 14x + 50 as one of the factors.
the other is (x+9)
so the polynomial could be
f(x) = (x+9)(x^2 + 14x + 50)
To form a third degree polynomial function with real coefficients given the zeros -7 + i and -9, we can use the concept of complex conjugates.
First, we know that if -7 + i is a zero, then its conjugate -7 - i must also be a zero. Similarly, if -9 is a zero, then its conjugate -9 must also be a zero.
So the factors of the polynomial can be expressed as:
(x - (-7 + i))(x - (-7 - i))(x - (-9))
Next, let's simplify each factor:
(x + 7 - i)(x + 7 + i)(x + 9)
Now, we can expand these factors:
(x + 7 - i)(x + 7 + i)(x + 9)
= (x^2 + 7x + ix + 7x + 49 + 7i - ix - 7i - i^2)(x + 9)
= (x^2 + 14x + 49 - i^2)(x + 9)
= (x^2 + 14x + 49 - (-1))(x + 9)
= (x^2 + 14x + 50)(x + 9)
Finally, multiplying the remaining factors, we obtain the third degree polynomial function with real coefficients:
f(x) = (x^2 + 14x + 50)(x + 9)
So, the third degree polynomial function with real coefficients that satisfies the given conditions is f(x) = (x^2 + 14x + 50)(x + 9).
To form a third-degree polynomial function with real coefficients, we need to consider the complex conjugate pairs of the zeros. Since -7 + i is a zero, its conjugate, -7 - i, must also be a zero. Therefore, our zeros are -7 + i, -7 - i, and -9.
To form the polynomial function, we can start by using the product of linear factors method. If a number is a zero of a polynomial, then (x - zero) is a factor of the polynomial. Therefore, we can write the factors for our zeros as follows:
(x - (-7 + i))(x - (-7 - i))(x - (-9)) = 0
Simplifying the expressions, we get:
(x + 7 - i)(x + 7 + i)(x + 9) = 0
Now, expanding the expression:
(x^2 + 7x + ix + 7x + 49 + 7i - ix - i^2)(x + 9) = 0
Since i^2 is equal to -1, we have:
(x^2 + 14x + 50)(x + 9) = 0
Expanding further:
x^3 + 14x^2 + 50x + 9x^2 + 126x + 450 = 0
Combining like terms:
x^3 + 23x^2 + 176x + 450 = 0
Therefore, the third-degree polynomial function with real coefficients whose zeros are -7 + i and -9 is:
f(x) = x^3 + 23x^2 + 176x + 450