y = (x-4)(x-7i)(x+7i)
now just expand that.
now just expand that.
So, we have the zeros: 4, 7i, and -7i. This means the factors of the polynomial will be (x-4), (x-7i), and (x+7i).
To get the polynomial function, we multiply all the factors together:
(x-4) * (x-7i) * (x+7i)
Expanding this expression:
(x - 4) * (x^2 - (7i)^2)
Simplifying further:
(x - 4) * (x^2 - 49i^2)
Since i^2 is equal to -1, we can simplify even more:
(x - 4) * (x^2 + 49)
Now, we can multiply the factors together:
(x - 4)(x^2) + (x - 4)(49)
(x^3 - 4x^2) + (49x - 196)
Finally, we can combine like terms to get the polynomial function in standard form:
f(x) = x^3 - 4x^2 + 49x - 196
Therefore, the polynomial function in standard form with real coefficients whose zeros include 4, 7i, and -7i is f(x) = x^3 - 4x^2 + 49x - 196.
Let's break this down step by step:
Step 1: Start by writing the factorization of the polynomial using the given zeros:
The polynomial will have three factors: (x - 4), (x - 7i), and (x + 7i).
(Note: Since 7i and -7i are complex conjugates, we can rewrite (x - 7i) as (x + 7i).)
Step 2: Multiply the factors together to find the polynomial:
Multiply (x - 4) with (x - 7i) and (x + 7i) to get:
(x - 4)(x - 7i)(x + 7i)
Step 3: Simplify the expression to find the polynomial function:
Using the difference of squares, we can simplify (x - 7i)(x + 7i) to (x^2 - (7i)^2) as follows:
(x^2 - (7i)^2) = (x^2 - 49i^2)
Since i^2 is equal to -1, we substitute -1 for i^2:
(x^2 - 49i^2) = (x^2 - 49(-1))
(x^2 - 49i^2) = (x^2 + 49)
Now, the polynomial function in standard form with real coefficients becomes:
(x - 4)(x^2 + 49)
Expanding the given factorization, we get:
(x - 4)(x^2 + 49) = x^3 - 4x^2 + 49x - 196
Therefore, the polynomial function in standard form with real coefficients whose zeros include 4, 7i, and -7i is:
f(x) = x^3 - 4x^2 + 49x - 196.