Find the polynomial function P of the lowest possible degree, having real coefficients, with the given zeros.
3+2i, -2 and 1
The zeroes of polynomials are either real or complex. Complex zeroes always come in with the conjugates.
Since the three zeroes above do not include conjugates, the minimum degree of polynomial is 3+1=4.
The polynomial having zeroes of
3+2i, -2, 1
and the conjugate 3-2i is therefore
P(x)=(x-(-2))(x-1)(x-3-2i)(x-3+2i)
=(x+2)(x-1)(x²-6x+13)
Expand if necessary
Well, finding a polynomial function with the given zeros is like assembling a puzzle. Let's break it down step by step with some "funny" equations.
First, the zero 3+2i tells us that the complex conjugate 3-2i must also be a zero. So, we can write two factors: (x - (3+2i)) and (x - (3-2i)).
Now, we have the zeros -2 and 1. These give us two more factors: (x - (-2)) and (x - 1).
To find the polynomial function P with the lowest degree, we can multiply all the factors together:
P(x) = (x - (3+2i))(x - (3-2i))(x - (-2))(x - 1)
Expanding this out, we get:
P(x) = (x^2 - 6x + 13)(x + 2)(x - 1)
To make sure we have real coefficients, we can multiply the complex factors:
P(x) = (x^2 - 6x + 13)(x^2 + x - 2)
Expanding this further, we have:
P(x) = x^4 - 5x^3 - 2x^2 + 7x + 26
And there you have it - a polynomial function with real coefficients and the given zeros. I hope my "funny" equations didn't make you feel like a clown!
To find a polynomial function with the given zeros, we can use the fact that if a number is a zero of a polynomial, then its conjugate is also a zero.
Here are the steps to find the polynomial function P:
Step 1: Start with the zeros: 3+2i, -2, and 1.
Step 2: Since 3+2i is a zero, its conjugate 3-2i is also a zero.
Step 3: Write the factors for the polynomial based on the zeros:
(x - (3+2i))(x - (3-2i))(x - (-2))(x - 1)
Step 4: Simplify the factors:
((x - 3 - 2i)(x - 3 + 2i))(x + 2)(x - 1)
Step 5: Expand and multiply the factors:
((x^2 - 6x + 9 + 4))(x + 2)(x - 1)
Step 6: Simplify further:
(x^2 - 6x + 13)(x + 2)(x - 1)
So, the polynomial function with the given zeros is P(x) = (x^2 - 6x + 13)(x + 2)(x - 1).
To find the polynomial function P with the given zeros, we can use the fact that complex zeros of a polynomial with real coefficients always come in conjugate pairs.
Given zeros:
1. 3 + 2i
2. -2
3. 1
Step 1: Since the given zero 3 + 2i is complex, its conjugate is also a zero of P. Therefore, the complex conjugate of 3 + 2i is 3 - 2i, which is the other complex zero.
Step 2: Write out the factors for the polynomial using the given zeros:
(x - (3 + 2i))(x - (3 - 2i))(x - (-2))(x - 1)
Step 3: Simplify the factors:
((x - 3) - 2i)((x - 3) + 2i)(x + 2)(x - 1)
Step 4: Multiply the factors:
((x - 3)^2 - (2i)^2)(x + 2)(x - 1)
((x - 3)^2 + 4)(x + 2)(x - 1)
Step 5: Expand and simplify the polynomial:
(x^2 - 6x + 9 + 4)(x + 2)(x - 1)
(x^2 - 6x + 13)(x + 2)(x - 1)
Therefore, the polynomial function P with the lowest possible degree and real coefficients, having the given zeros 3+2i, -2, and 1, is P(x) = (x^2 - 6x + 13)(x + 2)(x - 1).