2 polynomial qns. (1 Viewer)

[bos1234]

Write down the division identity when 30 divide 4 and 30 divide by 7

(b) Division of the polynomial P(x) by D9X) results in the quotient Q(x) and remainder R(x). Show that if P(x) is divided by Q(x), thew remainder will still b R(x). What is the quotient?

i dont understand the bold one.

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AND WHAT DOES HENCE DETERMINE THE HIGHEST MONIC COMMON FACTOR OF P(X),Q(X),R(X)

 

[jyu]

(b) Division of the polynomial P(x) by D9X) results in the quotient Q(x) and remainder R(x). Show that if P(x) is divided by Q(x), thew remainder will still b R(x). What is the quotient?

i dont understand the bold one.

P(x) divided by D(X) results in the quotient Q(x) and remainder R(x)

i.e. P(x)/D(x) = Q(x) + R(x)/D(x)

.: P(x) = D(x)Q(x) + R(x)

.: P(x)/Q(x) = D(x) + R(x)/Q(x)

i.e. P(x) divided by Q(X) results in the quotient D(x) and remainder R(x).

:wave:

 

[jyu]

i dont understand the bold one.

AND WHAT DOES HENCE DETERMINE THE HIGHEST MONIC COMMON FACTOR OF P(X),Q(X),R(X)

e.g. (2x - 1) = 2(x - 1/2)
(2x - 1) is non-monic, (x - 1/2) is monic.

(3x^2 + 6x - 5) = 3(x^2 + 2x - 5/3)
(3x^2 + 6x - 5) is non-monic, (x^2 + 2x - 5/3) is monic.

Example

P(x) = D(x)Q(x) + R(x)
P(x) = (x^4 + x^2 + 3)(4x^4 - 1) + (2x^3 + 2x^2 - x - 1)
=(x^4 + x^2 + 3)(2x^2 + 1)(2x^2 - 1) + (x + 1)(2x^2 - 1)
=(x^4 + x^2 + 3)(2x^2 + 1)(sqrt2 x - 1)(sqrt2 x + 1)
+ (x + 1)(sqrt2 x - 1)(sqrt2 x + 1)

(sqrt2 x - 1) and (sqrt2 x + 1) are common factors of Q(x), R(x) and consequently of P(x).

(2x^2 - 1) is the highest common factor of Q(x), R(x) and consequently of P(x).

(x^2 - 1/2) is the highest monic common factor of Q(x), R(x) and consequently of P(x).


:wave:

 

[ronaldinho]

e.g. (2x - 1) = 2(x - 1/2)
(2x - 1) is non-monic, (x - 1/2) is monic.

(3x^2 + 6x - 5) = 3(x^2 + 2x - 5/3)
(3x^2 + 6x - 5) is non-monic, (x^2 + 2x - 5/3) is monic.

Example

P(x) = D(x)Q(x) + R(x)
P(x) = (x^4 + x^2 + 3)(4x^4 - 1) + (2x^3 + 2x^2 - x - 1)
=(x^4 + x^2 + 3)(2x^2 + 1)(2x^2 - 1) + (x + 1)(2x^2 - 1)
=(x^4 + x^2 + 3)(2x^2 + 1)(sqrt2 x - 1)(sqrt2 x + 1)
+ (x + 1)(sqrt2 x - 1)(sqrt2 x + 1)

(sqrt2 x - 1) and (sqrt2 x + 1) are common factors of Q(x), R(x) and consequently of P(x).

(2x^2 - 1) is the highest common factor of Q(x), R(x) and consequently of P(x).

(x^2 - 1/2) is the highest monic common factor of Q(x), R(x) and consequently of P(x).


:wave:

can u plz show another example? an easier one?

thanks bye

 

[jyu]

can u plz show another example? an easier one?

thanks bye

May be you try this one:

P(x) = 4x^8 - x^6 + 4x^4 - 4x^3 + 3x^2 + x - 1 is divided by D(x) = x^4 + 1.

Find Q(x) and R(x).

Find the highest monic common factor of P(x), Q(x) and R(x).

:wave:

 

[bos1234]

Show that

x^2 -3x + 2 is a factor of P(x) = x^n(2^m-1) + x^m(1 - 2^n) + (2^n - 2^m)

 

[jyu]

Show that

x^2 -3x + 2 is a factor of P(x) = x^n(2^m-1) + x^m(1 - 2^n) + (2^n - 2^m)

Note that x^2 -3x + 2 =(x - 1)(x - 2)

Use the factor theorem to show (x - 1) and (x - 2) are factors of P(x).

.: x^2 -3x + 2 is a factor of P(x).


:wave: