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Polynomial (1 Viewer)

Lukybear

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Remainder of P(x) = x^3-3x^2 + 4x -2 when divided by (x+i)

Isnt P(-i) = -i + 3 -4i -2

since (-1 x i)^2 = 1 x i^2 = -1?
 

Lukybear

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Could you do it for me? Divide P(x) by (x+i) Just to make sure.
 

Lukybear

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One more question.

If P(x) is a monic, degree 4, integer coefficent and constant term of 2. P(x) has a zero i, and a rational zero. The sum of the zeroes of P(x) is a postive real number. Find P(x), factorise fully with R.

After solving question, we get ab = 2, where a and b are roots

The solution given states roots are 1,2 since 1+2 >0. However, can a and b be other values, that satisfy P(x), i.e. 1/2 and 4?
 

Lukybear

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If someone could help me to solve this question, I would be very much appreciated.

P(x) is a monic polynomial of degree 4 with integer coefficents and constant term 4. One Zero is sqrt 2, another zero is rational and the sum of the zeros is positive. Factorise P(x) fully over R
 

stampede

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One more question.

If P(x) is a monic, degree 4, integer coefficent and constant term of 2. P(x) has a zero i, and a rational zero. The sum of the zeroes of P(x) is a postive real number. Find P(x), factorise fully with R.

After solving question, we get ab = 2, where a and b are roots

The solution given states roots are 1,2 since 1+2 >0. However, can a and b be other values, that satisfy P(x), i.e. 1/2 and 4?
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mso-style-qformat:yes; mso-style-parent:""; margin-top:0cm; margin-right:0cm; margin-bottom:10.0pt; margin-left:0cm; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:Calibri; mso-fareast-theme-font:minor-latin; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi; mso-fareast-language:EN-US;} .MsoChpDefault {mso-style-type:export-only; mso-default-props:yes; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:Calibri; mso-fareast-theme-font:minor-latin; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi; mso-fareast-language:EN-US;} .MsoPapDefault {mso-style-type:export-only; margin-bottom:10.0pt; line-height:115%;} @page Section1 {size:595.3pt 841.9pt; margin:72.0pt 72.0pt 72.0pt 72.0pt; mso-header-margin:35.4pt; mso-footer-margin:35.4pt; mso-paper-source:0;} div.Section1 {page:Section1;} --> </style><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin-top:0cm; mso-para-margin-right:0cm; mso-para-margin-bottom:10.0pt; mso-para-margin-left:0cm; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi;} </style> <![endif]--> eqn in root form is:

x^4 - (α+β+γ+δ)x^3 + (αβ+αγ+αδ+βγ+βδ+γδ)x^2 - ......
where a, β, γ, δ are the roots.

sum of roots: α+β+γ+δ = α+β (since if i is a root, then -i is also a root, ie: α + β + i - i = α+β)

now, sum of roots = -b/a which has to be an integer since the q says so.
therefore α+β = -b/a = integer = k

therefore: α+β = k

if α and β are .5 and 4, they do not add up to an integer, k.
get it?
 

Lukybear

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2010
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font-family:"Calibri","sans-serif"; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:Calibri; mso-fareast-theme-font:minor-latin; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi; mso-fareast-language:EN-US;} .MsoChpDefault {mso-style-type:export-only; mso-default-props:yes; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:Calibri; mso-fareast-theme-font:minor-latin; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi; mso-fareast-language:EN-US;} .MsoPapDefault {mso-style-type:export-only; margin-bottom:10.0pt; line-height:115%;} @page Section1 {size:595.3pt 841.9pt; margin:72.0pt 72.0pt 72.0pt 72.0pt; mso-header-margin:35.4pt; mso-footer-margin:35.4pt; mso-paper-source:0;} div.Section1 {page:Section1;} --> </style><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin-top:0cm; mso-para-margin-right:0cm; mso-para-margin-bottom:10.0pt; mso-para-margin-left:0cm; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi;} </style> <![endif]--> eqn in root form is:

x^4 - (α+β+γ+δ)x^3 + (αβ+αγ+αδ+βγ+βδ+γδ)x^2 - ......
where a, β, γ, δ are the roots.

sum of roots: α+β+γ+δ = α+β (since if i is a root, then -i is also a root, ie: α + β + i - i = α+β)

now, sum of roots = -b/a which has to be an integer since the q says so.
therefore α+β = -b/a = integer = k

therefore: α+β = k

if α and β are .5 and 4, they do not add up to an integer, k.
get it?
But -b/a doesnt really have to be interger does it? It says a+b add to a no.>0 and only specify that b and a are integers. So b/a could be fraction
 

stampede

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But -b/a doesnt really have to be interger does it? It says a+b add to a no.>0 and only specify that b and a are integers. So b/a could be fraction

from ur q:

If P(x) is a monic, degree 4, integer coefficent

also, a is 1 since the eqn is monic
therefore -b/a is just -b
so alpha+beta = -b which has to be an integer
 

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