Polynomial Function with degree of 4 [closed]
The graph of a polynomial function of degree 4 is shown below.
Graph with real roots at $(-4,0)$ and $(2,0)$ and a $y$ intercept of $(0,-32)$
Explain why, of the four roots of the equation $f(x)=0$, two are real and two are complex.
I know that when there is a negative under a radical it produces complex roots, but I do not know how to explain why there are 2 real roots and 2 complex roots from just a graph.
algebra-precalculus
edited Dec 3 '18 at 8:56
DRF
4,477926
asked Dec 2 '18 at 23:56
elliana adeson
61
closed as unclear what you're asking by Jean-Claude Arbaut, Rory Daulton, Brahadeesh, ancientmathematician, DRF Dec 3 '18 at 8:56
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
The graph of a polynomial function of degree 4 is shown below.
Graph with real roots at $(-4,0)$ and $(2,0)$ and a $y$ intercept of $(0,-32)$
Explain why, of the four roots of the equation $f(x)=0$, two are real and two are complex.
I know that when there is a negative under a radical it produces complex roots, but I do not know how to explain why there are 2 real roots and 2 complex roots from just a graph.
algebra-precalculus
edited Dec 3 '18 at 8:56
DRF
4,477926
asked Dec 2 '18 at 23:56
elliana adeson
61
closed as unclear what you're asking by Jean-Claude Arbaut, Rory Daulton, Brahadeesh, ancientmathematician, DRF Dec 3 '18 at 8:56
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
3
Where is the graph? I ain't seen it yet.
– DeepSea
Dec 2 '18 at 23:59
You might want to add the graph. Also please use mathjax to format your questions. What do you know about polynomials? Have you learned the Fundamental theorem of Algebra? Please add information about what you have thought about.
– DRF
Dec 3 '18 at 8:38
add a comment |
The graph of a polynomial function of degree 4 is shown below.
Graph with real roots at $(-4,0)$ and $(2,0)$ and a $y$ intercept of $(0,-32)$
Explain why, of the four roots of the equation $f(x)=0$, two are real and two are complex.
I know that when there is a negative under a radical it produces complex roots, but I do not know how to explain why there are 2 real roots and 2 complex roots from just a graph.
algebra-precalculus
edited Dec 3 '18 at 8:56
DRF
4,477926
asked Dec 2 '18 at 23:56
elliana adeson
61
The graph of a polynomial function of degree 4 is shown below.
Graph with real roots at $(-4,0)$ and $(2,0)$ and a $y$ intercept of $(0,-32)$
Explain why, of the four roots of the equation $f(x)=0$, two are real and two are complex.
I know that when there is a negative under a radical it produces complex roots, but I do not know how to explain why there are 2 real roots and 2 complex roots from just a graph.
algebra-precalculus
algebra-precalculus
edited Dec 3 '18 at 8:56
DRF
4,477926
asked Dec 2 '18 at 23:56
elliana adeson
61
edited Dec 3 '18 at 8:56
DRF
4,477926
asked Dec 2 '18 at 23:56
elliana adeson
61
edited Dec 3 '18 at 8:56
DRF
4,477926
edited Dec 3 '18 at 8:56
DRF
4,477926
edited Dec 3 '18 at 8:56
DRF
4,477926
4,477926
asked Dec 2 '18 at 23:56
elliana adeson
61
asked Dec 2 '18 at 23:56
elliana adeson
61
asked Dec 2 '18 at 23:56
elliana adeson
61
61
closed as unclear what you're asking by Jean-Claude Arbaut, Rory Daulton, Brahadeesh, ancientmathematician, DRF Dec 3 '18 at 8:56
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as unclear what you're asking by Jean-Claude Arbaut, Rory Daulton, Brahadeesh, ancientmathematician, DRF Dec 3 '18 at 8:56
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
3
Where is the graph? I ain't seen it yet.
– DeepSea
Dec 2 '18 at 23:59
You might want to add the graph. Also please use mathjax to format your questions. What do you know about polynomials? Have you learned the Fundamental theorem of Algebra? Please add information about what you have thought about.
– DRF
Dec 3 '18 at 8:38
add a comment |
3
Where is the graph? I ain't seen it yet.
– DeepSea
Dec 2 '18 at 23:59
You might want to add the graph. Also please use mathjax to format your questions. What do you know about polynomials? Have you learned the Fundamental theorem of Algebra? Please add information about what you have thought about.
– DRF
Dec 3 '18 at 8:38
3
3
Where is the graph? I ain't seen it yet.
– DeepSea
Dec 2 '18 at 23:59
Where is the graph? I ain't seen it yet.
– DeepSea
Dec 2 '18 at 23:59
You might want to add the graph. Also please use mathjax to format your questions. What do you know about polynomials? Have you learned the Fundamental theorem of Algebra? Please add information about what you have thought about.
– DRF
Dec 3 '18 at 8:38
You might want to add the graph. Also please use mathjax to format your questions. What do you know about polynomials? Have you learned the Fundamental theorem of Algebra? Please add information about what you have thought about.
– DRF
Dec 3 '18 at 8:38
add a comment |
1 Answer
1
active
oldest
votes
A polynomial if degree $n$ will always have $n$ roots. If your graph of a 4th order polynomial shows 2 real roots, and you know that there are no more roots in the graph, then 2 are "missing". These two must be complex.
Also interesting is that if the coefficients of the polynomial are real then it's complex roots always appear in conjugate pairs. Therefore, again, if you know you have one complex root then you must have it's conjugate as well.
Algebraically, this is because the square root operation always produces a positive and a negative square root. For instance, $sqrt{4} = pm 2$ since either positive it negative 2, when squared, give you 4.
answered Dec 3 '18 at 0:02
Michael Stachowsky
1,260417
The graph shows two roots, but the graph window may simply too small to include two other roots. I.e. it may be that two roots are to the left or to the right of what the graph shows. So your first paragraph is not convincing.
– Rory Daulton
Dec 3 '18 at 1:07
Fair, but op had additional information. I'll try to edit the first paragraph
– Michael Stachowsky
Dec 3 '18 at 2:40
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
A polynomial if degree $n$ will always have $n$ roots. If your graph of a 4th order polynomial shows 2 real roots, and you know that there are no more roots in the graph, then 2 are "missing". These two must be complex.
Also interesting is that if the coefficients of the polynomial are real then it's complex roots always appear in conjugate pairs. Therefore, again, if you know you have one complex root then you must have it's conjugate as well.
Algebraically, this is because the square root operation always produces a positive and a negative square root. For instance, $sqrt{4} = pm 2$ since either positive it negative 2, when squared, give you 4.
answered Dec 3 '18 at 0:02
Michael Stachowsky
1,260417
The graph shows two roots, but the graph window may simply too small to include two other roots. I.e. it may be that two roots are to the left or to the right of what the graph shows. So your first paragraph is not convincing.
– Rory Daulton
Dec 3 '18 at 1:07
Fair, but op had additional information. I'll try to edit the first paragraph
– Michael Stachowsky
Dec 3 '18 at 2:40
add a comment |
A polynomial if degree $n$ will always have $n$ roots. If your graph of a 4th order polynomial shows 2 real roots, and you know that there are no more roots in the graph, then 2 are "missing". These two must be complex.
Also interesting is that if the coefficients of the polynomial are real then it's complex roots always appear in conjugate pairs. Therefore, again, if you know you have one complex root then you must have it's conjugate as well.
Algebraically, this is because the square root operation always produces a positive and a negative square root. For instance, $sqrt{4} = pm 2$ since either positive it negative 2, when squared, give you 4.
answered Dec 3 '18 at 0:02
Michael Stachowsky
1,260417
The graph shows two roots, but the graph window may simply too small to include two other roots. I.e. it may be that two roots are to the left or to the right of what the graph shows. So your first paragraph is not convincing.
– Rory Daulton
Dec 3 '18 at 1:07
Fair, but op had additional information. I'll try to edit the first paragraph
– Michael Stachowsky
Dec 3 '18 at 2:40
add a comment |
A polynomial if degree $n$ will always have $n$ roots. If your graph of a 4th order polynomial shows 2 real roots, and you know that there are no more roots in the graph, then 2 are "missing". These two must be complex.
Also interesting is that if the coefficients of the polynomial are real then it's complex roots always appear in conjugate pairs. Therefore, again, if you know you have one complex root then you must have it's conjugate as well.
Algebraically, this is because the square root operation always produces a positive and a negative square root. For instance, $sqrt{4} = pm 2$ since either positive it negative 2, when squared, give you 4.
answered Dec 3 '18 at 0:02
Michael Stachowsky
1,260417
A polynomial if degree $n$ will always have $n$ roots. If your graph of a 4th order polynomial shows 2 real roots, and you know that there are no more roots in the graph, then 2 are "missing". These two must be complex.
Also interesting is that if the coefficients of the polynomial are real then it's complex roots always appear in conjugate pairs. Therefore, again, if you know you have one complex root then you must have it's conjugate as well.
Algebraically, this is because the square root operation always produces a positive and a negative square root. For instance, $sqrt{4} = pm 2$ since either positive it negative 2, when squared, give you 4.
answered Dec 3 '18 at 0:02
Michael Stachowsky
1,260417
edited Dec 3 '18 at 2:41
answered Dec 3 '18 at 0:02
Michael Stachowsky
1,260417
answered Dec 3 '18 at 0:02
Michael Stachowsky
1,260417
answered Dec 3 '18 at 0:02
Michael Stachowsky
1,260417
1,260417
The graph shows two roots, but the graph window may simply too small to include two other roots. I.e. it may be that two roots are to the left or to the right of what the graph shows. So your first paragraph is not convincing.
– Rory Daulton
Dec 3 '18 at 1:07
Fair, but op had additional information. I'll try to edit the first paragraph
– Michael Stachowsky
Dec 3 '18 at 2:40
add a comment |
The graph shows two roots, but the graph window may simply too small to include two other roots. I.e. it may be that two roots are to the left or to the right of what the graph shows. So your first paragraph is not convincing.
– Rory Daulton
Dec 3 '18 at 1:07
Fair, but op had additional information. I'll try to edit the first paragraph
– Michael Stachowsky
Dec 3 '18 at 2:40
The graph shows two roots, but the graph window may simply too small to include two other roots. I.e. it may be that two roots are to the left or to the right of what the graph shows. So your first paragraph is not convincing.
– Rory Daulton
Dec 3 '18 at 1:07
The graph shows two roots, but the graph window may simply too small to include two other roots. I.e. it may be that two roots are to the left or to the right of what the graph shows. So your first paragraph is not convincing.
– Rory Daulton
Dec 3 '18 at 1:07
Fair, but op had additional information. I'll try to edit the first paragraph
– Michael Stachowsky
Dec 3 '18 at 2:40
Fair, but op had additional information. I'll try to edit the first paragraph
– Michael Stachowsky
Dec 3 '18 at 2:40
add a comment |
3
Where is the graph? I ain't seen it yet.
– DeepSea
Dec 2 '18 at 23:59
You might want to add the graph. Also please use mathjax to format your questions. What do you know about polynomials? Have you learned the Fundamental theorem of Algebra? Please add information about what you have thought about.
– DRF
Dec 3 '18 at 8:38