Determine whether the first polynomial can be expressed as a linear combination of other two polynomials.2x^3-2x^2+12x-6,x^3-2x^2-5x-3,3x^3-5x^2-4x-9

FobelloE 2021-09-03 Answered

Determine whether the first polynomial can be expressed as a linear combination of other two polynomials.
2x32x2+12x6,

x32x25x3,

3x35x24x9

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Expert Answer

timbalemX
Answered 2021-09-04 Author has 108 answers

The given polynomials are
2x32x2+12x6
x32x25x3
3x35x24x9
The polynomial can be written as vector form
v1=[22126],v2=[1253],v2=[3549]
Let, v1=av2+bv3
[22126]=a[1253]+b[3549]
[22126]=[a+3b2a5b5a4b3a9b]
a+3b=25a4b=12
2a5b=23a9b=6
using equation
6b5b=42
b=2
a=4
These values of a,b are also santisty, hence polynomial having linear relationship
P1=4P2+2P3
or 2P3=P1+4P2
P1= I polynomial
P2= II polynomial
P3= III polynomial

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