Write a polynomial of least degree with real coefficients and with the root
β5+13π
the other root is -5-13i, so
y = (x-(-5+3i))(x-(-5-13i))
y = ((x+5)-13i)((x+5)+13i)
y = (x+5)^2 + 13^2
y = x^2+10x+194
or
sum of roots = -5-13i + -5+13i = -10
product of roots = (-5-13i)(-5+13i)
= 25 - 169i^2 = 25+169 = 194
x^2 + 10x + 194
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To write a polynomial with the given complex root, we can make use of the conjugate pair theorem. The conjugate pair theorem states that if a polynomial has real coefficients, then complex roots must occur in conjugate pairs. In other words, if a+bi is a root of a polynomial with real coefficients, then a-bi must also be a root.
In this case, the given root is -5+13π. Since we know that complex roots occur in conjugate pairs, the conjugate of -5+13π is -5-13π. Therefore, -5-13π must also be a root of the polynomial.
To find the polynomial, we can set up the factors based on the roots:
(x - (-5 + 13π))(x - (-5 - 13π))
Simplifying this expression:
(x + 5 - 13π)(x + 5 + 13π)
Using the difference of squares formula (a^2 - b^2 = (a + b)(a - b)):
[(x + 5)^2 - (13π)^2]
Squaring binomials:
(x^2 + 10x + 25 - 169π^2)
The imaginary unit i^2 is defined as -1, so the expression simplifies further:
(x^2 + 10x + 25 + 169)
Combining like terms, the polynomial is:
x^2 + 10x + 194