EXERCISE -1
A) Which of the following algebraic expressions are polynomials?
1) x²+ 3x+ 4. Y
2) x²+ x √x. N
3) x²+ x √3 + 4. Y
4) x²+ 2√x + 4. N
5) (x²- 2x + 1)/(2x +3). N
6) 7x² - √3 x+9.
7) x³ + x² -1/x +5.
8) x²+ x √x +5.
9) (√3 x³+ 5x²+7x +9)/(x+2).
10) x² - 5x +9/x.
11) (x³+ 3x²++3x+1)/(x²+2x+1).
B) Which of the following expressions are rational expressions?
1) (x³- 3x²+2)/(x² +1). Y
2) (x³- 3x²)/(2x +3). Y
3) (x²-1)/(2 √x +3). N
4) (√3 x²- 4x +8)/(x -√8). Y
5) (x³+ 5x²+4)/(x² + 2 √x -1). N
6) (3x² - 5 √x+6)/(x -2). N
7) (x²-1)/(2 √x +5). N
8) (x²+x/√3 +5)/(4√x -3). N
9) (x³+x²-9x+4)/(2x+3).
10) (x²- 3x+5)/(x+1).
11) (x²+2 √x +5)/(x²+1).
12) (√3 x² -4x +12)/(√x -1).
13) (x³- 3x²+ 5 √x +9)/(x²+ x+1).
1) (x² - x -12)/(x² +8x +15). (x -4)/(x+5)
2) (x² +3x -10)/(x² -3x +2). (x +5)/(x-1)
3) (x² +2x - 3)/(x² - 2x -15). (x -1)/(x-5)
4) (6x² - 7x -20)/(9x² +6x -8). (2x -5)/(3x-2)
5) (x² - 7x +10)/(x² -3x -10). (x -2)/(x+2)
6) (2x² +3x +1)/(2x² +x -1). (2x +1)/(2x-1)
7) (x² +x -2)/(2x² +x -3). (x +2)/(2x+3)
8) (x² -5x -6)/(x² +3x +2). (x -6)/(x+2)
9) (x² +5x -6)/(x² +10x +24). (x -1)/(x+4)
10) (x² +7x +12)/(x² +4x +3). (x +4)/(x+1)
11) (x² -6x +8)/(x² -3x +2). (x -4)/(x-1)
12) (2x² - x -10)/(2x² -11x +15). (x +2)/(x-3)
13) (4x² - 13x +3)/(4x -1). (x-3)
14) (6x² - 5x +1)/(9x² +12x -5). (2x -1)/(3x+5)
15) (x-3)(x² - 5x +4)/(x-1)(x² -2x -3). (x -4)/(x+1)
16) (x-3)(x² - 5x +4)/(x-4)(x² -2x -3). (x -6)/(x+2)
17) (8x³ -125)/(4x² +10x +25). (2x -5)
18) (3x³ -24)/(2x² -15x +22). 3(x²+ 2x +4)/(2x-11)
19) (x -3)(x² - 5x +4)/(x-4)(x² -2x -3). (x -1)/(x+1)
20) (2x⁴- 162)/(x² +9)(2x -6). x+3
21) (x²-1)(x+2)(x² - x -72)/(x-9)(x +1). (x -1)(x+2)(x+8)
22) (x-1)(x-2)(x² - 9x +14)/(x -7)(x² -3x +2). (x -2)
23) (x+5)(x² +7x +10)/(x-3)(x² -3x +2). (x +5)²(x+2)/(x-1)(x-2)(x-3)
24) (5x² - 15x³)/(1- 9x²). 5x²/(3x+1)
25) 4(x² +x -1)/(4x² -4). (2x -1)/(x-1)
26) (x³ - 19x +30)/(x² -3x -40). (x -3)(x-2)/(x-8)
27) (x³ - 3x +2)/(2x³- 4x² +6x -4). (x+2)(x -1)/2(x²- x+2)
28) (x³- 2x² - x +2)/(x³- x² - 4x +4). (x +1)/(x+2)
29) (x³ +216)/(x⁴- 6x³+ 36x²). (x +6)/x²
30) (x³ - x -6)/(x³- 3x² +4). (x²+ 2x +3)/(x²- x -2)
31) (x² - 7x +6)/(x³+ 2x² -x -2). (x -6)/(x+1)(x+2)
32) (x³+1)/(x⁴ +x²+1). (x +1)/(x²+x+1)
33) (x² + 2x +4)/(x⁴+4x² +16). 1/(x²- 2x+ 4)
34) (x³+ 4x² +3x)/(x⁴- 10x² +9). x /(x-1)(x-3)
35) (x² - 4)(x²-8x +7)/(x -2)(x² -6x -7). (x²+x -2)/(x+1)
36) (x⁴- 13x² +36)/(x³- x² - 6x). (x+3)(x -2)/x
37) (2x⁴ - 2x)/(x³+ x² +x). 2(x - 1)
38) (x⁴- 10x² +9)/(x³+ 4x² +3x). (x -3)(x-1)/x
40) (x-4)(6x² +29x +35)/(2x+5)(3x² +19x +28). (x -4)/(x+4)
41) (x+5)(x² +7x +10)/(x-3)(x² +10x +25). (x +2)/(x-3)
42) (x³+ 9x² +20x)/(x² +5x +4). x(x +5)/(x+1)
43) (x(x² - 3x +2)/(x²y- 2xy). (x -1)/y
44) √[{(x²+3x+2)(x² +5x +6)}/{x²(x² +4x +3)}]. (x +2)/x
45) (x-1)(x -2)(x² - 9x +14)/(x-7)(x² -3x +2). (x -2)
46) (y-3)(y² - 5y +4)/(y-4)(y² -2y -3). (x-3)(x² - 5x +4)/(x-4)(x² -2x -3). (y -1)/(y+1) . (x -1)/(x+1)
47) (x³- x² + x -1)/(x³- x² - 3x +3). (x² +1)/(x²-3)
48) (x³- 6x² +11x -6)/(x³+ x² - 24x +36). (x -1)/(x+6)
49) (x³- x -6)/(x³- 3x² +4). (x²+ 2x +3)/(x²- x -2)
50) (x³- 8x² +19x -12)/(x³-3x² - x +3). (x -4)/(x+1)
52) (x⁴+ 8x³ +23x² +28x +12)/(x⁴+ 4x³+ 3 x²). (x²+ 4x +4)/x²
53) (x⁴+ 10x³ +35x² +50x +24)/(x⁴+ 4x³-7 x² -34x -24). (x +3)/(x -3)
54) (2x⁴+ 3x³ -12x² -7x +6)/(4x⁴+ 4x³- 25 x² - x +6). (x +2)/(2x+1)
55) (x³ -6x² +11x -6)/(x⁴+ 2x³-7x²+8x+12). (x -1)/(x²+3x +2)
56) (x⁴(a²+ b²)x² + a²b²)/(x⁴+ b⁴). (x²- a²)/(x²+ b²)
57) (x³ +8)/(x⁴+ 4x²+16). (x +3)/(x²+ 2x+4)
58) (x⁸-y⁸)/(x⁶- y⁶). (x² +y²)(x⁴+ y⁴)/(x²+ xy+ y²)(x²-xy+ y²)
59) (x² +x +1)/(x⁴+ x²+1). 1/(x²-x +1)
60) (x² +2x +2)/(x⁴+ 4x³+ 4). 1/(x²-2x +2)
61) (x³ +8)/(x⁴+ 4x²+16). (x+2)/(x²+2x +4)
62) (x³ +2x² -2x +3)/(x³-8x+3). (x²-x +1)/(x²-3x+1)
63) 4(x² +x -2)/6(x³+2x² - x -2). 2/3(x +1)
64) (3x³ -16x² +23x -6)/(2x³-11x²+17x-6). (3x -1)/(2x-1)
65) (2x³ -5x² +x +2)/(2x³+ 3x²- 3x -2). (x -2)/(x+2)
66) (4x⁴+ 12x³ +7x² -3x -2)/(x³+ 2x²- x -2). (4x²-1)/(x-1)
EXERCISE -2
1) (x+3)/(x+2) + (x+1)/(x- 2). 2(x²+ 2x- 2)/(x² - 4)
2) (x²+ x-1)/(x²-1) + (x+1)/(x³+ 2). (x⁵+ x⁴+ 3x²+ x- 3)/(x² - 1)(x³+2)
3) (x+1)/(x-1) + (x²-1)/(x+1). (x³+ x+ 2)/(x² - 1)
4) (x²+1)/(x+3) + (x²- 3)/(x +3). 2(x²- 1)/(x +3)
5) (x²- 3x)/(x+2) + (x+2)/(x² -3x).
6) (x+1)/(x-1) + (x-1)/(x+1). 2(x²+1)/(x² - 1)
7) 4x/(x²- 1) + (x+1)/(x- 1). (x²+ 6x +1)/(x² - 1)
8) (x² - 9)/(x+2) + (x³ - 27)/(x+ 2). (x³+ x²- 36)/(x + 2)
9) (3x+2)/(x² -16) + (x-5)/(x+ 4)². (4x²+ 5x+ 28)/(x - 4)(x+4)²
10) (x²+ x-1)/(x²-1) + (x+1)/(x³+ 2). (x⁵+ x⁴+ 3x²+ x- 3)/(x² - 1)(x³+2)
11) (x²- 4)/(x²+ 4x+4) - (x²- 2x-3)/(x+1). (-x²+ 2x+4)/(x +2)
12) (2x+1)/(2x-1) - (2x-1)/(2x+1). 8x/(4x²-1)
13) (x²- x- 6)/(x²-9) + (x²+ 2x-24)/(x²- x-12). 2(x+4)/(x +3)
14) (x+1)/(x- 1) + (x²- 1)/(x+1). (x²- x+2)/(x -1)
15) (a²+2a +3)/(a²-1) + (a- 4)/(a+1). (2a² - 3a+7)/(a² -1)
16) {(2x²+3)/(x-1) +(x+3)/(x+1)}+ x/(x² - 1). x(x²+ 3x+6)/(x² -1)
17) (2x³+ x²+3)/(x²+ 2)² - 1/(x²+2). (2x³+1)/(x² +2)²
18) {(x²+1)/(x-1) + (x+ 1)/(2x+1)} -{(x-1)/(x+2) + (x+1)/(x -2)} (2x⁵- 2x⁴- 4x³ - 14x²- 4x+4)/(x -1)(2x+1)(x²- 4)
19) 1/(1- x) + 1/(1+ x) + 2/(x²+ 1) + 4/(1+ x⁴). 8/(1 - x⁸)
20) 1/(x- 1) - 1/(x+1) - 2/(x²+ 1) - 4/(x⁴+1). 8/(x⁸ -1)
21) 1/(x²- 7x +12) + 1/(x²- 5x+6).
2/{(x- 2)(x- 4)}
22) 1/(x²- 5x+6) + 1//(x²- 3x+2) + 2/(x²- 8x+ 15). 4/(x²- 6x +5)
23) (x- 3)/(x²- x- 64) + (2x -1)/(2x²+ 5x-3) - (2x+5)/(x²+ 5x +6). 0
24) 1/(1+ x + x²) - 1/(1- x + x²) + 2x/(1+ x² + x⁴). 0
25) (x²- x - 6)/(x² - 9) + (x²+ 2x-24)/(x²- x- 12). 2(x+4)/(x +3)
26) x/(x- y) + y/(x +y) + 2xy/(y² - x²). 1
27) (1+ x)/(1- x) - (1- x)/(1+ x) + 4x/(1+ x²) + 8x³/(1- x⁴). 8x/(1- x²)
28) 1/(x+ a) + 1/(x - b) + 1/(x+ c) + ax/(x³ + ax²) + bx/(x³+ bx²) + cx/(x³+ cx²). 3/x
29) x/(x- y) - y/(x+ y) - 2xy/(x²- y²). (x-y)/(x+y)
30) {(x²+1)/(x -1) + (x+ 1)/(2x+1)} - { (x- 1)/(x +2) + (x+1)/(x-2). (2x⁵- 2x⁴- 4x³ - 14x² - 4x+4)/(x-1)(2x +1)(x+2)(x-2)
31) (x²+ 3x+2)/(x+4)+ (x-6)/(x+2)} - {(x³+ x²+2)/(x²+ 6x +8)+ (x² -8)/(x²+ 6x+8)}. 2(2x²+3x- 7)/(x +2)(x+4)
EXERCISE -3
1) (x+ 1)/(x +2) + (x²- 1)/(x²+1). (2x³+3x²-1)/(x+2)(x²+1)
2) (x+ 1)/(x -1)² + 1/(x+1). 2(x²+1) /x(x²-1)
3) (x+ 4)/(x +2) - (x- 1)/(x -2). (x-6)/(x²-4)
4) 4x/(x² -1) - (x+1)/(x-1). -(x-1)/(x+1)
5) 2x/(x²-1) + 3x/(x²-3x+2). (5x²-x)/(x-2)(x²-1)
6) {(x+ 1)/(x -1) + (x- 1)/(x+1)} - 3x²/(x-1). (-3x³-x²+2)/(x²-1)
7) (x+ 3)/(x -2) - (x+ 1)/(x-3). (x-7)/(x²-5x+6)
8) (x+ 2)/(x -3) - (x- 1)/x. 6x-3)/(x²-3x)
9) (x+ 1)/(x² -1) - 2/x. -(x-2)/x(x-1)
10) (4x - x²+ 1)/(1-x) - (2x+x²- 1)/(2+x). - 3(x²+2x+1)/(x²+x-2)
11) {(x+ 1)/(x -1) - (x- 1)/(x+1)} - 3x²/(x-1). (-3x³-3x²+4x)/(x²-1)
12) (x²+ 4)/(x -2) - (x²+2x- 6)/(x+2). 2(2x²+7x-2)/(x²-4)
13) (x²+x+ 1)/(x -3) - (2x+ 1)/(3-x). (x²+3x+2)/(x-3)
14) {(2x+ 5)/(x +1) + (x²+1)/(x²-1)} - (3x -2)/(x-1). 2/(x+1)
15) x³/(x- 1) - x²/(x +1) - 1/(x-1)+ 1/(x+1). (x²+2)
16) {(x²+3x+ 2)/(x +4) + (x- 6)/(x+2)} - {(x³+x²+2)/(x²+6x+8) + (x²-8)/(x²+6x+8)} 2(2x²+ 3x-7)/(x²+6x+8)
17) (1+ 8x) + (8x -1)/(1+8x) - (2+2x)/(5+2x). (128x³+352x²+102x-2) /(16x²+42x+5)
18) (x²-x-6)/(x² -9) + (x²+2x- 24)/(x²-x-12). (2x+8)/(x+3)
19) {(2x²+ 3)/(x -1) + (x+3)/(x+1)} + x/(x²-1). x(2x²+3x+6)/(x²-1)
20) (x-3)/(x²-7x +12) + ²(x- 1)/(x²-4x+3) - 3(x -4)/(x²- 5x+4). (7x-25)/(x³ - 6x²+19x-12)
21) {x²-(y-z)²}/{(x +z)²- y²} + {y²- (x -z)²}/{(x+y)²- z²} + {z²- (x-y)²}/{(y+z)²- x²}. 1
22) (1+x)/(1-x) - (1- x)/(1+x)+ 4x/(1+ x²) 8x/(1- x⁴)
EXERCISE -4
1) If A= (x+5)/(x-8) and B= (x-5)/(x+8), find A- B. 26x/(x²- 64)
2) If P= (x+1)/(x-1) and Q= (x-1)/(x+1), find P+ Q. 2(x²+1)/(x²- 1)
3) If A= (x²+1)/(x²-1) and B= (x+1)/(x+2), find A+ B. (2x³+3x²+1)/(x²- 1)(x+2)
4) If A= (x-1)/(x-2) and B= (x+1)/(x-3), find A+ B. 26x/(x²- 64)
5) If A= (x+2)/(x-2) and B= (x-2)/(x+2), find A+ B. 2(x²+4)/(x²- 4)
6) If A= (x+2)/(x-3) and B= (x-2)/(x- 3), find A- B. 2x/(x- 3)
7) If A= (x+1)/(x²-2) and B= (x²-2)/(x+1), find A+B. (x⁴-3x²+2x+5)/(x³+x²- 2x -2)
8) If P= 4x/(x²-1) and Q= (x+1)/(x-1), find P+ Q. (x²+6x+1)/(x²- 1)
9) If P= (x²-9)/(x+2) and Q= (x³-27)/(x+2) find P+ Q. (x³+x²-36)/(x+2)
10) If A= (3x+2)/(x²-16) and B= (x-5)/(x+4)², find A+ B. (4x²+5x+28)/(x- 4)(x+4)²
11) If A= (2√3 x²+1)/2x and B= -3√3 x² +1)/3x, find A+ B. 5/6x
12) If A= (x+1)/(x-1)² and B= 1/(x+1), find A+ B. 2(x²+1)/(x- )²(x+1)
13) If P= (x²-4)/(x²+4x+4) and Q= (x² - 2x-3)/(x+1), find P- Q. (-x²+2x+4)/(x+2)
14) If P= (2x+1)/(2x-1) and Q= (2x-1)/(2x+1), find P- Q. 8x/(4x²- 1)
EXERCISE -5
1) which rational expression should be added to (x³- 1)/(x²+2) to get (2x³- x²+3)/(x²+2). (x³ - x²+ +4)/(x² +2)
2) What should be substracted from the expression (5x³+x²- 1)/(x+ 2) to get (1- 2x+x³) ? (4x³- 2x² - x +1)/(x+2).
3) ax²/(x- a)(x- b)(x -c) +bx/(x- b)(x - c) + c/(x - c) + 1. x³/(x- a)( x- b)(x - c)
4)
1) (x²+ 3)/(x² -1) X (x- 1)/2x. (x²+3)/(2x²+2x)
2) 4x/(x² -1) X (x+1)/(x -1). 4x/(x -1)²
3) If A= x - 1/x then find the value of A - 1/A. (x⁴- 3x²+1)/(x³- x)
4) If A= (x+1)/(x -1), find A - 1/A. 4x/(x²- 1)
5) If A= 4x + 1/x, find A+ 1/A. (16x⁴+ 9x²+1)/(4x³+x)
6) If R= (x²+ 2)/(x - 3) and S= (x-1)/x , find (R( S)/(R X S). (x³+ x²- 2x+ 3)/(x³- x²+ 2x- 2)
7) (x²- 3x-4)/(x²- 9) X (x²- x- 6)/(x²+ 3x +2). (x-4)/(x+3)
8) (x³- 8)/(x² -4) X (x²+ 6x+ 8)/(x²- 2x+1). (x³+ 6x²+ 12x+16)/(x²- 2x +1)
9) {(2x²+3)/(x -1) + (x+3)/(x+1)} X (x²- 1)/3x. (2x² + 3x+5)/3
10) (2x²+3x+ 1)/(3x²+ 4x+1) X (4x²+ 5x+1)/(5x²+ 6x+1) X (15x²+ 8x+1)/(8x²+ 6x +1). 1
11) (2x² + x- 6)/(x²+ 4x- 5) X (x³- 3x²+ 2x)/(4x² - 6x) + (3 - x²)/( x²-25)
(x³- 7x²- 4x+26)/2(x² - 25)
12) x/{(x + y)²- 2xy} X x⁴ - y⁴)/{(x+y)³ - 3xy(x+y)} X {(x+y)²- 3xy}/{(x +y)² - 4xy}. x/(x - y)
13) [(1/x + 1/y) {x- y)²+ xy}] + [(1/x - 1/y) {(x+y)² - xy}]. 2y²/x
14) /(x²+ 4x+4) - (x²- 2x-3)/(x+1). (-x²+ 2x+4)/(x +2)
(-x²+ 2x+4)/(x +2)
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