Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.
3, -13, and 5 + 4i
Urgently need help
since complex roots come in pairs, we have
(x-3)(x+13)(x-(5+4i))(x-(5-4i))
(x-3)(x+13)((x-5)-4i)((x-5)+4i)
(x-3)(x+13)((x-5)^2+4^2)
(x^2+10x-39)(x^2-10x+41)
x^4 - 98x^2 + 800x - 1599
Thanks
write a polynomial function of minimum degree with real coefficients whose zeros include those listed. write the polynomial in standard form of -1 and 3+2i
Write the polynomial function P(X) WHOSE ZERO ARE 0,4 AND - 6
To find a polynomial function with the given zeros, we need to consider that if a complex number is a zero, then its conjugate is also a zero.
Given zeros: 3, -13, and 5 + 4i.
Since 5 + 4i is a zero, its conjugate 5 - 4i will also be a zero.
To find the polynomial function, we start by using the fact that if a number is a zero, then the binomial (x - zero) is a factor of the polynomial.
Using this concept, we can write the factors for the given zeros:
(x - 3), (x + 13), and (x - (5 + 4i)) = (x - 5 - 4i).
Since its conjugate is also a zero, we include its factor: (x - (5 - 4i)) = (x - 5 + 4i).
Now, we multiply all the factors together to obtain the polynomial:
(x - 3)(x + 13)(x - 5 - 4i)(x - 5 + 4i).
To simplify this further, we can use the conjugate pair property of complex numbers:
(x - 3)(x + 13)((x - 5)^2 - (4i)^2).
Next, we simplify the terms inside the parentheses:
(x - 3)(x + 13)((x - 5)^2 - 16i^2).
Since i^2 = -1, we can substitute -1 into the equation:
(x - 3)(x + 13)((x - 5)^2 - 16(-1)).
This simplifies to:
(x - 3)(x + 13)((x - 5)^2 + 16).
Finally, we can distribute and simplify further if needed:
(x^2 - 3x + 13x - 39)(x^2 - 10x + 25 + 16).
This simplifies to:
(x^2 + 10x - 39)(x^2 - 10x + 41).
Expanding further by multiplying both quadratics:
x^4 - 10x^3 + 41x^2 + 10x^3 - 100x^2 + 410x - 39x^2 + 390x - 1599.
Combining like terms:
x^4 - 139x^2 + 800x - 1599.
Therefore, the polynomial function of minimum degree with real coefficients and the given zeros is:
f(x) = x^4 - 139x^2 + 800x - 1599 (in standard form).