write a polynomial function in standard form with real coefficients whose zeros include 4, 7i, and -7i

Question

write a polynomial function in standard form with real coefficients whose zeros include 4, 7i, and -7i

Answer 1

To find a polynomial function with the given zeros, we can use the fact that complex zeros always come in conjugate pairs. In this case, since 7i is a zero, -7i must also be a zero.

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.

Answer 2

To find a polynomial function in standard form with real coefficients whose zeros include 4, 7i, and -7i, we know that complex zeros occur in conjugate pairs. This means that if 7i is a zero, then -7i must also be a zero.

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.

Answer 3

y = (x-4)(x-7i)(x+7i)

now just expand that.