For a polynomial f(x) with real coefficients having the given degree and seros.
Degree 4; zeros: 5+4 i; 5 multiplicity 2.
I need to write down the polynomial.
(Write an expression using x as the variable. Use integers or fractions for any numbers in the expression. Please simplify your answer.)
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You need to remember how to write the factored form of a polynomial when you know its zeroes:
`f(x) = a(x-x_1)(x-x_2)(x-x_3)(x-x_4)`
Notice that the degree of polynomial decides the number of roots, hence, if the poynomial is of fourth order, thus it has 4 roots.
Notice that the problem provides a complex root, hence, the conjugate of the complex root also will be a root to the given polynomial.
Since the complex root is `5 + 4i` , then the conjugate root is `5 - 4i` .
Notice that the problem provides the information that 5 is a root of multiplicity 2, hence 5 is the root of polynomial two times such that:
`f(x) = (x-(5 + 4i))(x-(5- 4i))(x-5)(x-5)`
`f(x) = (x - 5 - 4i)(x - 5 + 4i)(x - 5)^2`
You need to convert the product `(x - 5 - 4i)(x - 5 + 4i)` into a difference of squares such that:
`(x - 5 - 4i)(x - 5 + 4i) = (x - 5)^2 - (4i)^2`
`(x - 5 - 4i)(x - 5 + 4i) = (x - 5)^2+ 16`
Notice that `i^2 =-1` (complex number theory)
`f(x) = ((x - 5)^2 + 16)(x - 5)^2 `
`f(x) =(x - 5)^4 + 16(x - 5)^2`
Expanding the binomial yields:
`(x - 5)^2 = x^2 - 10x + 25`
`(x - 5)^4 = (x^2 - 10x + 25)^2 = x^4 + 100x^2 + 625 - 20x^3 + 50x^2 - 500x`
`f(x) = x^4 + 100x^2 + 625 - 20x^3 + 50x^2 - 500x + 16x^2 - 160x + 400`
`f(x) = x^4 - 20x^3 +166x^2 - 660x+ 1025`
Hence, evaluating the polynomial under the given conditions yields `f(x) = x^4 - 20x^3 + 166x^2 - 660x + 1025.`
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