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[SOLVED] Find a polynominal function f(x) of least degrees having only the real coeficents and zeros as given

karush

Well-known member
Jan 31, 2012
2,724
Find a polynominal function f(x) of least degrees having only the real coeficents and zeros as given

$$5+i \quad 4-i$$
ok I did this but don't think this is the final answer
$(x-(5+i))(x-(5-i))=x^2+26$
$(x-(4-i))(x-(4+i))=x^2-17$
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
You've got the right idea, but your quadratic products aren't correct. :)
 

karush

Well-known member
Jan 31, 2012
2,724
You've got the right idea, but your quadratic products aren't correct. :)
$$(x-(5+i))(x-(5-i))=x^2+26$$
$$(x-(4-i))(x-(4+i))=x^2+17$$

is this as far as can go?

or is it

(
x^2+26)(x^2+17)
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
You want to first get two quadratics as you've done, and then the quartic polynomial will be their product, but you don't have the correct quadratics yet. Check your multiplications...you should have \(x\) terms. :)
 

karush

Well-known member
Jan 31, 2012
2,724
You want to first get two quadratics as you've done, and then the quartic polynomial will be their product, but you don't have the correct quadratics yet. Check your multiplications...you should have \(x\) terms. :)

$$(x^2 - 10 x + 26)(x^2 - 8 x +17)=x^4-18x^3+123x^2-378x+442$$
 

Olinguito

Well-known member
Apr 22, 2018
251
Find a polynominal function f(x) of least degrees having only the real coeficents and zeros as given

$$5+i \quad 4-i$$
ok I did this but don't think this is the final answer
$(x-(5+i))(x-(5-i))=x^2+26$
$(x-(4-i))(x-(4+i))=x^2-17$
A polynomial with real coefficients will have complex zeroes occurring in conjugate pairs. So if two of the zeroes are $5+i$ and $4-i$, the other ones will be $5-i$ and $4+i$. Hence $f(x)$ will be a polynomial of degree at least 4.

A polynomial with zeroes $a\pm bi$ is $x^2-2ax+(a^2+b^2)$. Hence
$$f(x)\ =\ k(x^2-10x+26)(x^2-8x+17)$$
where $k$ is any nonzero real number.
 

karush

Well-known member
Jan 31, 2012
2,724
What are some good online calculators for hunting down zeros

I thot EMath was nice and you
Pull latex from it.