22. If x3
+ ax2
+ bx + 6 has (x – 2) as a factor and leaves a remainder 3 when divided by (x – 3), find
the values of a and b.
23. Find the value of x3
+ y3
+ 15xy – 125 if x + y = 5.
24. Without actually calculating, find the value of (25)3 – (75)3
+ (50)3
.
25. Factorise each of the following cubic expressions:
(i) 8x3 – y
3 – 12x2
y + 6xy2
(ii) 27q3 – 125p3 – 135q2
p + 225qp2
(iii) 8x3 + 729 + 108x2 + 486x
(iv) 3 2 1 9 1 27
216 2 4
x x x - - +
26. Factorise:
(i) x3 + 216y3 + 8z3 – 36xyz
(ii) a3 – 64b3 – 27c3 – 36abc
27. Factorise: ( )
3 3 1 1 3
3 3 3 3
2 2
x y y z z x
æ ö æ ö ç ÷ ç ÷ - + - + - è ø è ø
28. Give one example each of a binomial of degree 35, and of a monomial of degree 100.
29. Find a zero of the polynomial p(x) = 2x + 1.
30. Verify whether 2 and 0 are zeroes of the polynomial x
2 – 2x.
31. Find the zero of the polynomial in each of the following cases:
(i) p(x) = x + 5 (ii) p(x) = x – 5 (iii) p(x) = 2x + 5
(iv) p(x) = 3x – 2 (v) p(x) = 3x (vi) p(x) = ax, a ¹ 0
32. Find the value of each of the following polynomials at the indicated value of variables:
(i) p(x) = 5x
2 – 3x + 7 at x = 1.
(ii) q(y) = 3y
3 – 4y + 11 at y = 2.
(iii) p(t) = 4t
4 + 5t
3 – t
2 + 6 at t = a.
33. Divide p(x) by g(x), where p(x) = x + 3x
2 – 1 and g(x) = 1 + x.
34. Divide the polynomial 3x
4 – 4x
3 – 3x –1 by x – 1.
35. Find the remainder obtained on dividing p(x) = x
3 + 1 by x + 1.
36. Find the remainder when x
4 + x
3 – 2x
2 + x + 1 is divided by x – 1.
37. Check whether the polynomial q(t) = 4t
3 + 4t
2 – t – 1 is a multiple of 2t + 1.
38. Check whether p(x) is a multiple of g(x) or not, where p(x) = x3 – x + 1, g(x) = 2 – 3x.
39. Check whether g(x) is a factor of p(x) or not, where p(x) = 8x3 – 6x2 – 4x + 3, g(x) = 1
3 4
x - .
40. Find the remainder when x
3 – ax
2 + 6x – a is divided by x – a.
41. Examine whether x + 2 is a factor of x
3 + 3x
2 + 5x + 6 and of 2x + 4.
Answers
Given : x³ + ax² + bx + 6 has (x – 2) as a factor and leaves a remainder 3 when divided by (x – 3)
To Find : values of a and b.
Solution:
x³ + ax² + bx + 6 has (x – 2) as a factor
=>for x = 2 value = 0
=> 2³ + a.2² + b.2 + 6 = 0
=> 8 + 4a + 2b + 6 = 0
=> 4a + 2b = - 14
=> 2a + b = - 7
leaves a remainder 3 when divided by (x – 3)
=> for x = 3 value = 3
=> 3³ + a.3² + b.3 + 6 = 3
=> 27 + 9a + 3b + 6 = 3
=> 9a + 3b = -30
=> 3a + b = -10
3a + b = -10
2a + b = - 7
=> a = - 3
b = - 1
Value of a = - 3
Value of b = -1
please post Questions one by one
Learn More:
on dividing x3-3x2+x+2 by a polynomial g(x),the quotient and ...
brainly.in/question/3135153
Divide 9x^3+3x^2-5x+7 by (3x-1) write the quotient and remainder ...
brainly.in/question/6190809
Divide the polynomial (6x³ + 11x² - 10x - 7) by the binomial (2x + 1 ...
brainly.in/question/4743553