Previous Exercise – Polynomials – Exercise 8.4 – Class X
Polynomials – Exercise 8.5
- Find the quotients and remainder using synthetic division.
(i) (x3 + x2 – 3x + 5) /(x – 1)
(ii) (3x3 – 2x2 + 7x – 5) /(x + 3)
(iii) (4x3 – 16x2 – 9x – 36)/(x + 2)
(iv) (6x4 – 29x3 + 40x2 – 12)/(x – 3)
(v) (8x4 – 27x2 + 6x + 9)/(x + 1)
(vi) (3x3 – 4x2 – 10x + 6)/(3x – 2)
(vii) (8x4 – 27x2 + 6x – 5)/(4x + 1)
(viii) (2x4 – 7x3 – 13x2 + 63x – 48)/(2x – 1)
- If the quotient obtained on dividng (x4 + 10x3 + 35x2 + 50x + 29) by (x + 4) is (x3 – ax2 + bx + 6) then find a, b and also the remainder
- If the quotient obtained on dividing (8x4 – 2x2 + 6x – 7) by (2x +1) is (4x3 + px2 – qx + 3) then find p, q and also find the remainder.
Polynomials – Exercise 8.5 – Solutions:
- Find the quotients and remainder using synthetic division.
(i) (x3 + x2 – 3x + 5) /(x – 1)
Solution:
Therefore, q(x) = x2 + 2x – 1 , r(x) = 4
(ii) (3x3 – 2x2 + 7x – 5) /(x + 3)
Solution:
Therefore, q(x) = 3x2 – 11x + 40 and r(x) = -125
(iii) (4x3 – 16x2 – 9x – 36)/(x + 2)
Solution:
Therefore, q(x) = 4x2 – 24x + 39 and r(x) = -114
(iv) (6x4 – 29x3 + 40x2 – 12)/(x – 3)
Solution:
Therefore, q(x) = 6x3 – 11x2 + 7x + 21 and r(x) = 51
(v) (8x4 – 27x2 + 6x + 9)/(x + 1)
Solution:
Therefore, q(x) = 8x3 – 8x2 – 19x + 25 and r(x) = -16
(vi) (3x3 – 4x2 – 10x + 6)/(3x – 2)
Solution:
Therefore, q(x) = 3x2 – 2x – 34/3 and r(x) = –14/9
⇒q(x) = x2 – 2/3 x – 34/9 and r(x) = –14/9
(vii) (8x4 – 2x2 + 6x – 5)/(4x + 1)
Solution:
Therefore, q(x) = 8x3 – 2x2 – 3/2x + 51/8 and r(x) = –211/32
⇒q(x) = 2x3 – 1/2x2 – 3/8 x + 51/32 and r(x) = –211/32
(viii) (2x4 – 7x3 – 13x2 + 63x – 48)/(2x – 1)
Solution:
Therefore, q(x) = 2x3 – 6x2 – 16x + 55 and r(x) = –41/2
⇒q(x) = x3 – 3x2 – 8x + 55/2 and r(x) = –41/2
- If the quotient obtained on dividing (x4 + 10x3 + 35x2 + 50x + 29) by (x + 4) is (x3 – ax2 + bx + 6) then find a, b and also the remainder
Solution:
Given, if the quotient obtained on dividing x4 + 10x3 + 35x2 + 50x + 29 by (x + 4) is (x3 – ax2 + bx + 6).
q’(x) = x3 – ax2 + bx + 6
On dividing x4 + 10x3 + 35x2 + 50x + 29 by (x + 4) quotient is x3 + 6x2 + 11x + 6 and the remainder is 5.
i.e., q(x) = x3 + 6x2 + 11x + 6 and r(x) = 5
q’(x) = q(x)
x3 + 6x2 + 11x + 6 = x3 – ax2 + bx + 6
Coefficient of x2 on both the sides, 6 = -a ⇒ a = -6
Coefficient of x on both the sides, 11 = b
Therefore, a = -6 and b = 11.
The remainder r(x) = 5.
- If the quotient obtained on dividing (8x4 – 2x2 + 6x – 7) by (2x +1) is (4x3 + px2 – qx + 3) then find p, q and also find the remainder.
Solution:
Given, if the quotient obtained on dividing (8x4 – 2x2 + 6x – 7) by (2x +1) is (4x3 + px2 – qx + 3).
q’(x) = (4x3 + px2 – qx + 3)
On dividing (8x4 – 2x2 + 6x – 7) by (2x +1) quotient is 8x3 – 4x2 + 6 and the remainder is -10
i.e., q(x) = 8x3 – 4x2 + 6 = 4x3 – 2x2 + 6 and r(x) = -10
q’(x) = q(x)
4x3 + px2 – qx + 3 = 4x3 – 2x2 + 6
Coefficient of x2 on both the sides, p = -2
Coefficient of x on both the sides, -q = 0 ⇒ q = 0
Therefore, p = -2 , q = 0 and r(x) = -10
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