Find a polynomial with real coefficients that has the given zeros.
-1 and 4-5i
(x+1)(x-(4-5i))(x-(4+5i))
now just expand that
Why did the polynomial go to the circus? Because it wanted to find some roots and have a real good time! 🎪🤡
To find a polynomial with the zeros -1 and 4-5i (where i is the imaginary unit), remember that complex zeros come in conjugate pairs. So, if 4-5i is a zero, then 4+5i must also be a zero.
So, our polynomial will have the factors (x - (-1))(x - (4-5i))(x - (4+5i)). Let's simplify that:
(x + 1)(x - (4-5i))(x - (4+5i))
Now, we can expand and simplify:
(x + 1)((x - 4) + 5i)((x - 4) - 5i)
(x + 1)((x - 4) - 5i)((x - 4) + 5i)
(x + 1)((x - 4)^2 - (5i)^2)
(x + 1)((x - 4)^2 - 25)
(x + 1)(x^2 - 8x + 16 - 25)
(x + 1)(x^2 - 8x - 9)
And there you have it! A polynomial with real coefficients that has the zeros -1 and 4-5i is (x + 1)(x^2 - 8x - 9). Enjoy the circus of algebra! 🎪🤡
To find a polynomial with real coefficients that has the given zeros, we need to consider the complex conjugates of the complex zeros as well. In this case, the complex conjugate of 4-5i is 4+5i.
The polynomial can be found by multiplying the factors corresponding to each zero.
If -1 is a zero, then (x + 1) is a factor of the polynomial.
If 4-5i is a zero, then (x - (4-5i)) is a factor of the polynomial.
Similarly, if 4+5i is a zero, then (x - (4+5i)) is a factor of the polynomial.
The polynomial can be written as:
(x + 1)(x - (4-5i))(x - (4+5i))
Simplifying this expression:
(x + 1)(x - 4 + 5i)(x - 4 - 5i)
Expanding the product of the factors:
(x + 1)((x - 4) + 5i)((x - 4) - 5i)
Using the difference of squares formula [(a - b)(a + b) = a^2 - b^2], the expression can be further simplified to:
(x + 1)((x - 4)^2 - (5i)^2)
(x + 1)(x^2 - 8x + 16 - 25i^2)
Notice that i^2 = -1, so (-25i^2) becomes 25.
(x + 1)(x^2 - 8x + 16 + 25)
Finally, expanding the product of the remaining factors:
(x + 1)(x^2 - 8x + 41)
This is a polynomial with real coefficients that has -1 and 4-5i as zeros.
To find a polynomial with real coefficients that has the given zeros, we use the fact that complex zeros occur in conjugate pairs. This means that if -1 + 0i is a zero, then -1 - 0i (which is just -1) is also a zero. Similarly, if 4 - 5i is a zero, then 4 + 5i is also a zero.
To construct the polynomial, we start by expressing the factors corresponding to the zeros.
For -1 + 0i, the factor is (x - (-1)) which simplifies to (x + 1).
For 4 - 5i, the factor is (x - (4 - 5i)) which simplifies to (x - 4 + 5i).
Since the zeros occur in conjugate pairs, the corresponding factors also occur in conjugate pairs. So, the factor for 4 + 5i is (x - 4 - 5i).
To obtain the polynomial, multiply all the factors together:
(x + 1) * (x - 4 + 5i) * (x - 4 - 5i)
Expanding and simplifying this expression will give us the polynomial.