Finding real coefficients of polynomial with complex roots
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I have been tasked with finding the just real coefficients of a polynomial with the roots:
$z_0=10$
$z_1=3-i$
$z_2=-8+2i$
Writing the polynomial as factors...
$(z-10)(z-3+i)(z+8-2i)$
..obviously does not work since this results in complex coefficients.
Can anyone help me out here?
algebra-precalculus polynomials complex-numbers
add a comment |Â
up vote
3
down vote
favorite
I have been tasked with finding the just real coefficients of a polynomial with the roots:
$z_0=10$
$z_1=3-i$
$z_2=-8+2i$
Writing the polynomial as factors...
$(z-10)(z-3+i)(z+8-2i)$
..obviously does not work since this results in complex coefficients.
Can anyone help me out here?
algebra-precalculus polynomials complex-numbers
âÂÂthe real coefficients of a polynomialâ or âÂÂa polynomial with only real coefficientsâÂÂ?
â Martin R
8 hours ago
Only real coefficients.
â Boris Grunwald
8 hours ago
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
I have been tasked with finding the just real coefficients of a polynomial with the roots:
$z_0=10$
$z_1=3-i$
$z_2=-8+2i$
Writing the polynomial as factors...
$(z-10)(z-3+i)(z+8-2i)$
..obviously does not work since this results in complex coefficients.
Can anyone help me out here?
algebra-precalculus polynomials complex-numbers
I have been tasked with finding the just real coefficients of a polynomial with the roots:
$z_0=10$
$z_1=3-i$
$z_2=-8+2i$
Writing the polynomial as factors...
$(z-10)(z-3+i)(z+8-2i)$
..obviously does not work since this results in complex coefficients.
Can anyone help me out here?
algebra-precalculus polynomials complex-numbers
algebra-precalculus polynomials complex-numbers
edited 8 hours ago
greedoid
31.1k94287
31.1k94287
asked 8 hours ago
Boris Grunwald
877
877
âÂÂthe real coefficients of a polynomialâ or âÂÂa polynomial with only real coefficientsâÂÂ?
â Martin R
8 hours ago
Only real coefficients.
â Boris Grunwald
8 hours ago
add a comment |Â
âÂÂthe real coefficients of a polynomialâ or âÂÂa polynomial with only real coefficientsâÂÂ?
â Martin R
8 hours ago
Only real coefficients.
â Boris Grunwald
8 hours ago
âÂÂthe real coefficients of a polynomialâ or âÂÂa polynomial with only real coefficientsâÂÂ?
â Martin R
8 hours ago
âÂÂthe real coefficients of a polynomialâ or âÂÂa polynomial with only real coefficientsâÂÂ?
â Martin R
8 hours ago
Only real coefficients.
â Boris Grunwald
8 hours ago
Only real coefficients.
â Boris Grunwald
8 hours ago
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
5
down vote
accepted
There is not enough information, say the degree of a given polynomial.
Anyway, since it has (only) real coeficients if $a+bi$ is one root, then $a-bi$ is also a root of the polynomial.
So this polynomial is divisible with:
$$q(z)=(z-10)colorred(z-3+i)(z-3-i)colorblue(z+8-2i)(z+8+2i) $$
$$= (z-10)(z^2-6z+10)(z^2+16z+68)$$
Thanks, I think this makes sense.
â Boris Grunwald
8 hours ago
And sorry, it was the smallest possible degree
â Boris Grunwald
8 hours ago
add a comment |Â
up vote
3
down vote
Polynomial with only real coefficients is
$$( z-10) , ( z-2i+8),(z+2 i+8) , ( z-i-3) , ( z+i-3)\=
z^5-118 z^3-68 z^2+3160 z-6800
$$
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
accepted
There is not enough information, say the degree of a given polynomial.
Anyway, since it has (only) real coeficients if $a+bi$ is one root, then $a-bi$ is also a root of the polynomial.
So this polynomial is divisible with:
$$q(z)=(z-10)colorred(z-3+i)(z-3-i)colorblue(z+8-2i)(z+8+2i) $$
$$= (z-10)(z^2-6z+10)(z^2+16z+68)$$
Thanks, I think this makes sense.
â Boris Grunwald
8 hours ago
And sorry, it was the smallest possible degree
â Boris Grunwald
8 hours ago
add a comment |Â
up vote
5
down vote
accepted
There is not enough information, say the degree of a given polynomial.
Anyway, since it has (only) real coeficients if $a+bi$ is one root, then $a-bi$ is also a root of the polynomial.
So this polynomial is divisible with:
$$q(z)=(z-10)colorred(z-3+i)(z-3-i)colorblue(z+8-2i)(z+8+2i) $$
$$= (z-10)(z^2-6z+10)(z^2+16z+68)$$
Thanks, I think this makes sense.
â Boris Grunwald
8 hours ago
And sorry, it was the smallest possible degree
â Boris Grunwald
8 hours ago
add a comment |Â
up vote
5
down vote
accepted
up vote
5
down vote
accepted
There is not enough information, say the degree of a given polynomial.
Anyway, since it has (only) real coeficients if $a+bi$ is one root, then $a-bi$ is also a root of the polynomial.
So this polynomial is divisible with:
$$q(z)=(z-10)colorred(z-3+i)(z-3-i)colorblue(z+8-2i)(z+8+2i) $$
$$= (z-10)(z^2-6z+10)(z^2+16z+68)$$
There is not enough information, say the degree of a given polynomial.
Anyway, since it has (only) real coeficients if $a+bi$ is one root, then $a-bi$ is also a root of the polynomial.
So this polynomial is divisible with:
$$q(z)=(z-10)colorred(z-3+i)(z-3-i)colorblue(z+8-2i)(z+8+2i) $$
$$= (z-10)(z^2-6z+10)(z^2+16z+68)$$
edited 8 hours ago
answered 8 hours ago
greedoid
31.1k94287
31.1k94287
Thanks, I think this makes sense.
â Boris Grunwald
8 hours ago
And sorry, it was the smallest possible degree
â Boris Grunwald
8 hours ago
add a comment |Â
Thanks, I think this makes sense.
â Boris Grunwald
8 hours ago
And sorry, it was the smallest possible degree
â Boris Grunwald
8 hours ago
Thanks, I think this makes sense.
â Boris Grunwald
8 hours ago
Thanks, I think this makes sense.
â Boris Grunwald
8 hours ago
And sorry, it was the smallest possible degree
â Boris Grunwald
8 hours ago
And sorry, it was the smallest possible degree
â Boris Grunwald
8 hours ago
add a comment |Â
up vote
3
down vote
Polynomial with only real coefficients is
$$( z-10) , ( z-2i+8),(z+2 i+8) , ( z-i-3) , ( z+i-3)\=
z^5-118 z^3-68 z^2+3160 z-6800
$$
add a comment |Â
up vote
3
down vote
Polynomial with only real coefficients is
$$( z-10) , ( z-2i+8),(z+2 i+8) , ( z-i-3) , ( z+i-3)\=
z^5-118 z^3-68 z^2+3160 z-6800
$$
add a comment |Â
up vote
3
down vote
up vote
3
down vote
Polynomial with only real coefficients is
$$( z-10) , ( z-2i+8),(z+2 i+8) , ( z-i-3) , ( z+i-3)\=
z^5-118 z^3-68 z^2+3160 z-6800
$$
Polynomial with only real coefficients is
$$( z-10) , ( z-2i+8),(z+2 i+8) , ( z-i-3) , ( z+i-3)\=
z^5-118 z^3-68 z^2+3160 z-6800
$$
answered 8 hours ago
Aleksas Domarkas
5154
5154
add a comment |Â
add a comment |Â
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âÂÂthe real coefficients of a polynomialâ or âÂÂa polynomial with only real coefficientsâÂÂ?
â Martin R
8 hours ago
Only real coefficients.
â Boris Grunwald
8 hours ago