4. Check wil
Using remainder theorem, find the remainder when p(X) is divided by q(X) in each
of the following:
() p(x) = x + 2x - 3x -x-1.9(x) = x - 2
(ii) p(x) = 4x' +12° +11 x -2,9(x) = 3x - 1
(ii) p(x) = 9x - 3x + x - 5,q(x) = 3x - 2
Answers
Answer:
We know that the division algorithm states that:
Dividend=(Divisor×quotient)+Remainder
Here it is given that the dividend is p(x)=4x
3
+2x
2
−10x+2, the divisor is g(x), the quotient is 2x
2
+4x+1 and the remainder is 5, therefore,
4x
3
+2x
2
−10x+2=[g(x)(2x
2
+4x+1)]+5
⇒g(x)=
2x
2
+4x+1
4x
3
+2x
2
−10x+2
The division is shown above.
Hence, from the above division, we get that the divisor is g(x)=2x−3
Answer:
(a) Q(X) =0
X +2 = 0
X = -2
P(X) = 2X³-3X²+4X-1
P(-2) = 2 × (-2)³ -3 × (-2)² + 4 × -2 -1
=> 2 × -8 - 3 × 4 -8-1
=> -16 -12-1 = -29
when P(X) divided by G(X) then reminder comes -29.
(B) X³-6X²-2X-4
Q(X) = 0
1-3X = 0
3X = 1
X = 1/3
P(X) = X³-6X²-2X-4
P(1/3) = (1/3)³ - 6 × (1/3)² - 2 × 1/3 -4
=> 1/27 - 6 × 1/9 -2/3 -4
=> 1/27 - 2/3 -2/3 -4
=> 1 - 18 - 18 - 108
=> -108 -36+1 = .-144+1 = -143.
When P(X) Divided by G(X) then reminder comes 73.
(C) 3X²+X-2
=> 3X²+3X-2X-2
=> 3X(X+1) -2(X+1)
=> (X+1) (3X-2)
=> (X+1) = 0. OR (3X-2) = 0
=> X = -1 OR X= 2/3
-1 and 2/3 are the two zeros of the polynomial 3X²+X-2.