Write a polynomial of least degree with real coefficients and with the root -6-8i. Write your answer using the variable x and in standard form with a leading coefficient of 1.
Thanks, I have no idea!
complex roots come in conjugate pairs, so if -6-8i is a root, so is -6+8i.
With those two roots, the least degree polynomial will just be degree 2, so we have
y = (x-(-6-8i))(x-(-6+8i))
= (x+6+8i)(x+6-8i)
= (x+6)^2 - (8i)^2
= x^2+12x+36+64
= x^2+12x+100
check here:
http://www.wolframalpha.com/input/?i=x^2%2B12x%2B100
scroll down a bit to where it shows the roots.
To find a polynomial with the root -6-8i, we need to remember that complex roots always come in conjugate pairs. Therefore, the conjugate of -6-8i is -6+8i.
To find the polynomial, we can use the fact that the factors of the polynomial are (x - root) for each root. So, the factors for our polynomial are (x - (-6-8i)) and (x - (-6+8i)).
Expanding these factors, we get:
(x + 6+8i) and (x + 6-8i).
To find the polynomial, we multiply these two factors together:
(x + 6+8i)(x + 6-8i)
= x(x + 6-8i) + 6(x + 6-8i) + 8i(x + 6-8i)
= x^2 + 6x - 8ix + 6x + 36 - 48i + 8ix + 48i - 64i^2
= x^2 + 12x + 36 - 64i^2
Since i^2 is equal to -1, we can simplify the polynomial further:
= x^2 + 12x + 36 + 64
= x^2 + 12x + 100
Therefore, the polynomial of least degree with real coefficients and with the root -6-8i is x^2 + 12x + 100.
To write a polynomial with -6-8i as a root, we need to consider its conjugate as well. Since the coefficients of the polynomial must be real, the conjugate of -6-8i will also be a root. The conjugate of -6-8i is -6+8i.
To find the polynomial, we can start by using the fact that if r is a root of a polynomial, then (x-r) is a factor of that polynomial.
Using this, we have:
(x - (-6-8i))(x - (-6+8i)) = (x + 6 + 8i)(x + 6 - 8i)
= (x + 6)(x + 6) - (8i)^2
= (x + 6)(x + 6) - 64i^2
= (x + 6)(x + 6) - 64(-1)
= (x + 6)(x + 6) + 64
= (x^2 + 12x + 36) + 64
= x^2 + 12x + 36 + 64
= x^2 + 12x + 100
So, the polynomial of least degree with real coefficients and with the root -6-8i is x^2 + 12x + 100.