Find a polynomial function of lowest degree with rational coefficients that has the given numbers as some of its zeros. 1 + i, 2

Wotzdorfg

Wotzdorfg

Answered question

2021-05-23

Find a polynomial function of lowest degree with rational coefficients that has the given numbers as some of its zeros.
1 + i, 2

Answer & Explanation

toroztatG

toroztatG

Skilled2021-05-24Added 98 answers

Complex zeros come in conjugate pairs so we know that 1-i is the third zero given that 1+i is a zero.
If a is a zero of a polynomial function, then x-a is one of its factor. Using 2, 1+i, and 1-i, the least possible degree of the polynomial function is 3:
f(x)=a(x2)[x(1+i)][x(1i)]
By expanging, f(x)=a(x2)[x2(1i)x(1+i)x+(1+i)(1i)]
f(x)=a(x2)[x2x+ixxix+(1i2)]
f(x)=a(x2)(x22x+2)
f(x)=a[x2(x2)2x(x2)+2(x2)]
f(x)=a(x32x2+4x+2x4)
f(x)=a(x34x2+6x4)
For simplicity, let a=1: f(x)=x34x2+6x4
Jeffrey Jordon

Jeffrey Jordon

Expert2021-08-11Added 2605 answers

Answer is given below (on video)

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