Algebraic Fraction
Exercise 11.1
1. Simplify:
(a) 3a² / 4a³= 3 × a × a / 4 × a × a × a
= 3 / 4a
(b) 5x²y / 10xy²= (5 × x × x × y) / (10 × x × y × y)
= (5 / 10) × (x / x) × (1 / y)
= x / 2y
(c) (a² + 2ab + b²) / (a² - b²)= (a + b) (a + b) / (a + b) (a - b)
= (a + b) / (a - b)
(d) (5a³ - 45a) / (4a² - 12a)= 5a (a² - 9) / 4a (a - 3)
= 5a (a - 3) (a + 3) / (4a(a - 3)
= 5 (a + 3) / 4
(e) (x - 3)³ / (2x - 6)= (x - 3) (x - 3) (x - 3) / 2(x - 3)
= (x - 3) (x - 3) / 2
= (x - 3)² / 2
(f) (x² + 6x + 9) / (x² - 9)= (x + 3) (x + 3) / (x + 3) (x - 3)
= (x + 3) / (x - 3)
(g) (a² + 6a + 8) / (a² - 16)= (a + 2) (a + 4) / (a + 4) (a - 4)
= (a + 2) / (a - 4)
(h) (x² + x - 12) / (x² - x - 6)= (x + 4) (x - 3) / (x - 3) (x + 2)
= (x + 4) / (x + 2)
(i) (2x + 3)² / (4x² - 9)= (2x + 3) (2x + 3) / (2x + 3) (2x - 3)
= (2x + 3) / (2x - 3)
(j) (x² + 2x - 15) / (x² + 9x + 20)= (x + 5) (x - 3) / (x + 5) (x + 4)
= (x - 3) / (x + 4)
(k) (x² + 5x + 6) / (x + 3)²= (x + 3) (x + 2) / (x + 3) (x + 3)
= (x + 2) / (x + 3)
(l) (x² - 9x + 18) / (x² - 7x + 6)= (x - 3) (x - 6) / (x - 1) (x - 6)
= (x - 3) / (x - 1)
(m) (x² - 1) / (x² - 6x + 5)= (x - 1) (x + 1) / (x - 1) (x - 5)
= (x + 1) / (x - 5)
(n) 3xyz / (3x² - 12x)= 3xyz / 3x (x - 4)
= yz / (x - 4)
(o) (x² - 4x + 4) / (x - 2)= (x - 2) (x - 2) / (x - 2)
= (x - 2)
Exercise 11.2 1. Simplify:(a) 2a / y + 3b / y
= (2a + 3b) / y
(b) 5p / r² + 7q / r²= (5p + 7q)/x²
(c) 5a / 6b - a / 6b
= (5a - a) / 6b = 4a / 6b = 2a / 3b
(d) 3 / (x + 2) - 2 / (x + 2)
= (3 - 2) / (x + 2) = 1 / (x + 2)
(e) (x + 1) / 2 + (x + 2) / 2
= (x + 1) + (x + 2) / 2 = (2x + 3) / 2
(f) (x + y) / (a + 1) - y / (a + 1)
= (x + y - y) / (a + 1) = x / (a + 1)
(g) 6 / (y - 3) - 3y / (y - 3)
= (6 - 3y) / (y -3 )
(h) 3x / (x + 1) + 3 / (x+1)
= (3x + 3) / (x + 1) = 3 (x + 1) / (x + 1) = 3
(i) mn / (m + n) - mn / (m + n)
= mn - mn / (m + n)
= 0
2. Simplify:(a) (x + 2) / (x + 3) + (x - 2) / (x + 3)
= (x + 2 + x - 2) / (x + 3) = 2x / (x + 3)
(b) (3x + 1) / (x² + 2) - (x + 1) / (x² + 2)
= (3x + 1 - (x + 1) / (x² + 2) = (3x + 1 - x - 1) / (x² + 2) = 2x / (x² + 2)
(c) y - 15) / (y² - 9) + 18 / (y² - 9)
= (y - 15 + 18) / (y² - 9)
= (y + 3) / (y + 3) (y - 3)
= 1 / (y - 3)
(d) (ax² + bx) / (x + a) + c/(x + a)
= (ax² + bx + c/(x + a)
= (ax² + bx + c)/(x + a)
(e) x² - 4x / (x² - 4) - 4 / (x² - 4)
Solution:
= (x² - 4x) / (x² - 4) - 4 / (x² - 4)
= (x² - 4x + 4) / (x² - 4)
= (x² - 2x - 2x + 4) / (x² - 4)
= x (x - 2) - 2 (x - 2) / (x² - 4)
= (x - 2) (x - 2) / (x + 2) (x - 2)
= (x - 2) /(x + 2)
(f) (y² + 3y) / (y + 3) + (5y + 15) / (y + 3)
Solution:
= (y² + 3y + 5y + 15) / (y + 3)
= y (y + 3) + 5 (y + 3) / (y + 3)
= (y + 3) (y + 5) / (y + 3)
= (y + 5)
(g) 5p² / (4 - p) - (35p - 60) / (4 - p)
Solution:
= 5p² - (35p - 60) / (4 - p)
= (5p² - 35p + 60) / (4 - p)
= (5p² - 20p - 15p + 60) / (4 - p)
= 5p (p - 4) - 15 (p - 4) / (4 - p)
= (p - 4) (5p + 15) / (4 - p)
= - (4 + P) (15 - 5p) / - (p + 4)
= 15 - 5p
(h) p⁴ / (p + 3)² + (81 - 18p²) / (p + 3)²
Solution:
= (p⁴ + 81 - 18p²) / (p + 3)²
= (p⁴ - 18p² + 81) / (p + 3)²
= (p⁴ - 9p² - 9p² + 81) / (p + 3)²
= p (p² - 9) - 9 (p² - 9) / (p + 3)²
= (p + 9) (p + 9) (p - 9) (p - 9) / (p + 3) (p + 3)
= (p - 9) (p - 9)
(i) 3x² / (x + y) + (6xy + 3y²) / (x + y)
Solution:
= (3x² + 6xy + 3y²) / (x + y)
= 3 (x² + 2xy + y²) / (x + y)
= 3 (x + y) (x + y) / (x + y)
= 3 (x + y)
(j) (a² + b²) / (a - b)²) - 2ab / (a - b)²
Solution:
= (a² + b²) / (a - b)²) - (2ab) / (a - b)²
= (a² + b² - (2ab) / (a - b)²
= (a - b)² / (a - b)² = 1
(k) m² / (m² + 5m + 6) + 2m / (m² + 5m + 6)
Solution:
= m² / (m² + 5m + 6) + 2m / (m² + 5m + 6)
= m² + 2m / (m² + 5m + 6)
= m (m + 2) / (m + 2) (m + 3)
= m / (m + 3)
(l) x² /(x² - 4x + 3) - 3x / (x² - 4x + 3)
Solution:
= x² / (x² - 4x + 3) - 3x / (x² - 4x + 3)
= (x² - 3x) / (x² - 4x + 3)
= x (x - 3) / (x - 3) (x - 1)
= x / (x - 1)
Exercise 11.3
1. Simplify:
(a) a / 3 + a / 4
Solution:
= a / 3 + a / 4
= (4a + 3a) / 12
= 7a / 12
(b) 2 / x + 3 / 2x
Solution:
= 2 / x + 3 / 2x
= (4 + 3) / 2x
= 7 / 2x
(c) 1 / 2a - 1 / 3a
Solution:
= (3 - 2) / 6a
= 1 / 6a
(d) 3x / 2y - 2x / 3y
Solution:
= (9x - 4x) / 6y
= 5x / 6y
(e) 2 / ab + 3 / 2bc
Solution:
= 2 / ab + 3 / 2bc
= (4c + 3a) / 2abc
(f) 4 / ax - 3 / bx
Solution:
= (4b - 3a) / abx
= (4b - 3a) / abx
(g) 4x + 3x / 7
Solution:
= (28x + 3x) / 7
= 31x / 7
(h) x² / 4 + y² / 3
Solution:
= (3x² + 4y²) / 12
= (3x² + 4y²) / 12
(i) 2 / a - 3 / ab
Solution:
= (2b - 3) / ab
= (2b - 3) / ab
(j) 3 / 7 - 5 / 3y
Solution:
= (9y - 35) / 21y
= (9y - 35) / 21y
(k) x² / y - 4y
Solution:
= (x² - 4y²) / y
= (x - 2y) (x + 2y) / y
(l) x / (2-x) - (2-x) / x
Solution:
= (x² - (2-x)²) / x (2-x)
= (x² - 4 + 4x - x²) / x (2-x)
= (4x - 4) / x (2-x)
= 4 (x - 1) / x (2-x)
2. Simplify:
(a) 2 / (x - y) + 3 / (x+y)
Solution:
= 2 (x + y) + 3 (x - y) / (x - y) (x + y)
= (2x + 2y + 3x - 3y) / (x - y) (x + y)
= (5x - y) / (x-y) (x+y)
(b) 1 / (a-b) - 1 / (a+b)
Solution:
= (a + b) - (a - b) / (a - b) (a + b)
= 2b / (a - b) (a + b)
= 2b / (a² - b²)
(c) 2 / (p - 2q) + 1 / (p+2q)
Solution:
= 2 (p + 2q) + 1 (p - 2q) / (p - 2q) (p + 2q)
= (2p + 4q + p - 2q) / (p - 2q) (p + 2q)
= (3p + 2q) / (p² - 4q²)
(d) x / 2 (x - 2) - 1 / (x - 2)
Solution:
= (x - 2) / 2 (x - 2)
= 1/2
(e) a / (a + b) + b / (a - b)
Solution:
= a (a - b) + b (a + b) / (a + b) (a - b)
= (a² - ab + ab + b²) / (a² - b²)
= (a² + b²) / (a² - b²)
(f) 3 / (x-a) + 4 / (x+a)
Solution:
= 3 (x+a) + 4 (x-a) / (x-a) (x+a)
= (3x + 3a + 4x - 4a) / (x²-a²)
= (7x - a) / (x² - a²)
(g) x / (x² - 1) + 1 / (x - 1)
Solution:
= x / (x - 1) (x + 1) + 1 / (x - 1)
= (x + x + 1) / (x - 1) (x + 1)
= (x + 1) / (x² - 1)
(h) (x + 3) / (x - 5) - (x + 5) / (x - 3)
Solution:
= (x + 3) (x - 3) - (x + 5) (x - 5) / (x - 5)(x - 3)
= (x² - 9 - (x² - 25) / (x² - 8x + 15)
= 16 / (x² - 8x + 15)
(i) (x + 7) / (x - 7) - x /(7 - x)
Solution:
Note: (7 - x) = - (x - 7)
= (x + 7) / (x - 7) + x / (x - 7)
= (x + 7 + x) / (x - 7)
= (2x + 7) / (x - 7)
(j) (2x + 1) / 6 + 2x
Solution:
= (2x + 1) / 6 + (12x) / 6
= (14x + 1) / 6
(k) x / 2 (x + y) - 2/ 3 (x + y)
Solution:
= (3x - 4) / 6(x + y)
(l) 1 / (x + 6) - x / (x + 9)
Solution:
= (x + 9) - x (x + 6) / (x + 6) (x + 9)
= (-x² + 3x + 9) / (x + 6) (x + 9)
(m) (x + 2) / (x² + x) - 3 / (x² - x - 2)
Solution:
= (x + 2) / x (x + 1) - 3 / (x - 1) (x + 2)
= (x + 2) (x - 1) - 3x (x + 1) / x (x + 1)(x - 1)
= (-2x - 2) / x (x + 1) (x - 1)
(n) 1/(x -3) + (3x - 5)/(x² -5x + 6)
Solution:
= 1 / (x - 3) + (3x - 5)/ (x - 3) (x - 2)
= (x - 2) + (3x - 5) / (x - 3) (x - 2)
= (4x - 7) / (x - 3) (x - 2)
(o) (2x - 1) / (x² + 4x) - (x - 2) / (x² + 2x - 8)
Solution:
Note: Factorize denominators: x² + 4x = x (x + 4) and x² + 2x - 8 = (x + 4) (x-2).
= (2x - 1) / x (x + 4) - (x -2) / (x + 4) (x - 2)
= (2x-1) (x-2) - x (x -2) / x (x+4) (x-2)
= (2x² - 4x - x + 2 - x² + 2x) / x (x+4) (x-2)
= (x² - 3x + 2) / x (x+4) (x-2)
= ((x-1) (x-2)) / x (x+4) (x-2)
= (x-1) / x (x+4)
(p) 2a / (a-1) - (a²+3) / (a²-1)
Solution:
2a / (a-1) - (a²+3) / (a²-1)
Note: Factorize denominator: a²-1 = (a-1)(a+1).
= 2a /(a-1) - (a²+3) / (a-1) (a+1)
= 2a (a+1) - (a²+3) / (a-1) (a+1)
= (2a² + 2a - a² - 3) / (a-1) (a+1)
= (a² + 2a - 3) / (a-1) (a+1)
= (a+3) (a-1) / (a-1) (a+1)
= (a+3) / (a+1)
(q) (a²+b²) / (a²-b²) - (a-b) / (a+b)
Solution:
Note: Factorize denominator: a²-b² = (a-b) (a+b).
(a²+b²) / (a²-b²) - (a-b) / (a+b)
= (a²+b²) / (a-b) (a+b) - (a-b) / (a+b)
= (a²+b²) - (a-b)² / (a-b) (a+b)
= {(a²+b² - (a² - 2ab + b²)} / (a-b) (a+b)
= 2ab / (a-b) (a+b)
= 2ab / (a²-b²)
(r) a / (a²+3a+2) - 2 / (a²-1)
Solution:
Note: Factorize denominators: a²+3a+2 = (a+1) (a+2) and a²-1 = (a-1) (a+1).
= a / (a+1) (a+2) - 2/ (a-1) (a+1)
= a (a-1) - 2 (a+2) / (a+1) (a+2) (a-1))
= (a² - a - 2a - 4) / (a+1) (a+2) (a-1)
= (a² - 3a - 4) / (a+1) (a+2) (a-1)
= (a-4) (a+1) / (a+1) (a+2) (a-1)
= (a-4) / (a+2) (a-1)
Exercise 11.4
1. Simplify:Exercise 11.4
1. Simplify:
(a) a²/b × 2/b
Solution:
= a² / b × 2 / b
= a × a / b × 2 / b
= a × a × 2 / b × b
= 2a² / b²
(b) 3x²/4y² × 4y/3x
Solution:
= 3x² / 4y² × 4y / 3x
= (3 × x × x) / (4 × y × y) × (4 × y) / (3 × x)
= (3 × 4 × x² × y) / (3 × 4 × y² × x)
= x / y
(c) 7a²b/8c × 4c²/14ab²
Solution:
= 7a²b / 8c × 4c² / 14ab²
= (7 × 4 × a² × b × c²) / (8 × 14 × c × a × b²)
= (28a²bc²) / (112abc)
= a / 4b
(d) x-y/x+y × x/y
Solution:
= (x - y) / (x + y) × x / y
= x(x - y) / y(x + y)
= x(x - y) / y(x + y)
(e) a-3/3 × 6/a-3
Solution:
= (a - 3) / 3 × 6 / (a - 3)
= 6(a - 3) / 3(a - 3)
= 6 / 3
= 2
(f) x-3/x+2 × (x+2)²/(x-3)²
Solution:
= (x - 3) / (x + 2) × (x + 2)² / (x - 3)²
= (x - 3) × (x + 2)² / [(x + 2) × (x - 3)²]
= (x + 2) / (x - 3)
2. Simplify:
(a) a²/b² ÷ a/b
Solution:
= a²/b² ÷ a/b
= a²/b² × b/a
= a × a × b / b × b × a
= a / b
(b) 3xy/4ab ÷ 6y/5b
Solution:
= (3xy / 4ab) ÷ (6y / 5b)
= (3xy / 4ab) × (5b / 6y)
= (3 × 5 × x × y × b) / (4 × 6 × a × b × y)
= 15x / 24a
= 5x / 8a
(c) x/7 ÷ x²/14
Solution:
= (x / 7) ÷ (x² / 14)
= (x / 7) × (14 / x²)
= (x × 14) / (7 × x²)
= 14 / (7x)
= 2 / x
(d) 6a²b/7x²y ÷ 6ab²/7y²
Solution:
= (6a²b / 7x²y) ÷ (6ab² / 7y²)
= (6a²b / 7x²y) × (7y² / 6ab²)
= (6 × 7 × a² × b × y²) / (7 × 6 × x² × y × a × b²)
= a × y / (x² × b)
= ay / (x²b)
(e) a² - b² / a ÷ a - b / b
Solution:
= (a² - b²) / a ÷ (a - b) / b
= (a + b) (a - b) / a × b / (a - b)
= (a + b) × b / a
= b (a + b) / a
(f) x² - 1 / y² ÷ x - 1 / y
Solution:
= (x² - 1) / y² ÷ [(x - 1) / y
= (x - 1) (x + 1) / y² × y / (x - 1)
= (x + 1) × y / y²
= (x + 1) / y
3. Simplify:
(a) x² - y² / x + y × x + y / (x - y)²
Solution:
= (x - y) (x + y) / (x + y) × (x + y) / (x - y) (x-y)
= (x + y) / (x - y)
(b) x² + 2xy + y² / x² - y² × x - y / x + y
Solution:
= (x + y)² / (x - y) (x + y) × (x - y) / (x + y)
= (x + y) (x + y) / (x - y) (x + y) × (x - y) / (x + y)
= 1
(c) (x² - 4x + 4) / (3y - xy) × (4x - 12) / (x - 2)
Solution:
= (x² - 4x + 4) / (3y - xy) × (4x - 12) / (x - 2)
= (x - 2)² / y (3 - x) × - 4 (3 - x) / (x - 2)
= (x - 2) (x - 2) / y (3 - x) × - 4 / (x - 2)
= -4 (x - 2) / y
(d) (a² - b²) / (a² + 2a + ab + 2b) × (a + 2) / (a + 3)
Solution:
= (a² - b²) / (a² + 2a + ab + 2b) × (a + 2) / (a + 3)
= (a - b) (a + b) / (a + 2)(a + b) × (a + 2) / (a + 3)
= (a - b)/(a + 3)
(e) (y² + 10y + 24) / (y² + 2y - 8) × (y - 3) / (y + 6)
Solution:
(y² + 10y + 24) / (y² + 2y - 8) × (y - 3) / (y + 6)
= (y + 6) (y + 4) / (y + 4)(y - 2) × (y - 3) / (y + 6)
= (y - 3) / (y - 2)
(f) (x² - 3x - 10) / (x² - 5x + 6) × (bx - 3b) / (cx - 5c)
Solution:
(x² - 3x - 10) / (x² - 5x + 6) × (bx - 3b) / (cx - 5c)
= (x - 5) (x + 2) / (x - 3) (x - 2) × b (x - 3) / c (x - 5)
= b(x + 2)/c(x - 2)
(g) (x² - 11x + 30)/(x² - 7x + 10) × (5x - 10)/(x² - 8x + 12)
Solution:
(x² - 11x + 30)/(x² - 7x + 10) × (5x - 10)/(x² - 8x + 12)
= (x - 5)(x - 6)/ (x - 5)(x - 2) × 5(x - 2)/ (x - 6)(x - 2)
= 5/(x - 2)
(h) (x² - 9)/(x² + 4x) × (x² + 2x - 8)/(x² + x - 6)
Solution:
(x² - 9)/(x² + 4x) × (x² + 2x - 8)/(x² + x - 6)
= (x - 3) (x + 3) / x (x + 4) × (x - 2) (x + 4)/ (x + 3) (x - 2)
= (x - 3) / x
(i) (x² - 5x + 6)/(x² - 6x + 9) × (x² - 2x - 3)/(x² - 3x + 2)
Solution:
(x² - 5x + 6)/(x² - 6x + 9) × (x² - 2x - 3)/(x² - 3x + 2)
= (x - 3)(x - 2)/ (x - 3)(x - 3) × (x - 3)(x + 1)/ (x - 2)(x - 1)
= (x + 1)
Division Problems:
4. Simplify:
(a) (x² - y²)/(x + y) ÷ (x - y)/(x + y)= (x² - y²)/(x + y) × (x + y)/(x - y)
= (x² - y²)/(x - y)
= (x + y)(x - y)/(x - y)
= x + y
(b) (x² - 5x + 6)/(x² - 9) ÷ (x - 3)/(x + 3)= (x² - 5x + 6)/(x² - 9) × (x + 3)/(x - 3)
= (x - 2)(x - 3)/(x - 3)(x + 3) × (x + 3)/(x - 3
= (x - 2)/(x - 3)
(c) (x² + 12x + 36)/(x² - 16) ÷ (3x + 18)/(2x² + 8x)= (x + 6)²/(x - 4)(x + 4) ÷ 3(x + 6)/(2x(x + 4)
= (x + 6)²/(x - 4)(x + 4) × 2x(x + 4)/3(x + 6)
= 2x(x + 6)/ 3(x - 4)
(d) (3x² - 4x - 7)/(3x² - 7x) ÷ (x² - 1)/(x - 4)= (3x - 7) (x + 1) / x (3x - 7) × (x - 4)/(x + 1) (x - 1)
= (3x - 7) (x + 1) / x (3x - 7) × (x - 4)/(x + 1) (x - 1)
= (x - 4)/ x (x - 1) (e) (x² + 2x - 15)/(x - 2) ÷ 3(x² + 4x - 5)/(x² - 3x + 2)
= (x + 5)(x - 3)/(x - 2) ÷ 3 (x + 5) (x - 1)/ x (x - 2)
= (x + 5)(x - 3) / (x - 2) × (x - 2) (x - 1) / 3 (x + 5) (x - 1))
= (x - 3) / 3
(f) (x² + 12x + 27) / (x² + x - 6) ÷ (x² + 4x - 45)/ 9 (x² - 4x - 5))= (x + 3) (x + 9) / (x - 2) (x + 3) ÷ (x + 9) (x - 5) / 9 (x - 5) (x + 1))
= (x + 3) (x + 9) / (x - 2) (x + 3) × 9 (x - 5) (x + 1) / (x + 9) (x - 5)
= 9 (x + 1) / (x - 2)
(g) (xy - x + 2y - 2) / (3y + 2x + xy + 6) ÷ (xy - x + 5y - 5) / (x² + 8x + 15)= x(y - 1) + 2 (y - 1) / x (y + 3) + 2 (y + 3) ÷ x(y - 1) + 5 (y - 1) / (x + 5) (x + 3)
= (y - 1) (x + 2) / (y + 2) (x + 3) ÷ (y - 1) (x + 5) /(x + 5) (x + 3)
= (y - 1) (x + 2) / (y + 2) (x + 2) x (x + 5) (x + 3) / (y - 1) (x + 5)
= (x + 2) / (y + 2)
(h) (y² + 4y - 12) / (y² - 5y + 6) ÷ (y² + 3y - 18) / (y² - 9)= (y + 6) (y - 2) / (y - 2) (y - 3) ÷ (y + 6) (y - 3) / (y + 3) (y - 3)
= (y + 6) (y - 2) / (y - 2) (y - 3) x (y + 3) (y - 3) / (y + 6) (y - 3)
= (y + 3) / (y - 3)
(i) (x² - 8x + 15) / (x² - 14x + 45) ÷ (x² - 2x - 15) / (x² - 8x - 9)= (x - 3)(x - 5)/(x - 9)(x - 5) ÷ (x - 5) (x + 3) / (x - 9) (x + 1)
= (x - 3)(x - 5)/(x - 9)(x - 5) x (x - 9) (x + 1) / (x - 5) (x + 3)
= (x - 3) (x + 1) / (x - 5) (x + 3)
(j) (a² + 3a + 2) / (a² - 4a - 12) ÷ (a² - a - 6) / (a² - 9a + 18)= (a + 1) (a + 2) / (a - 6) (a + 2) ÷ (a - 3) (a + 2) / (a - 6) (a - 3)
= (a + 1) (a + 2) / (a - 6) (a + 2) x (a - 6) (a - 3) / (a - 3) (a + 2)
= (a + 1)/(a + 2)
5. Simplify:= 2x / 5y × 2y / 5 × 3 / y)
= 2x x 2y x 3 / 5y x 5y
= 12x / 25y
(b) x / (x-1) - 1 / (x+1) ÷ (x-1) / (x²-1)= x (x + 1) - (x - 1) / (x - 1) (x + 1) ÷ (x - 1) / (x - 1) (x + 1)
= (x² + x - x + 1) / (x - 1) (x + 1) ÷ (x - 1) / (x - 1) (x + 1)
= (x² + 1) / (x - 1) (x + 1) × (x - 1) (x + 1) / (x - 1)
= (x² + 1) / (x - 1)
(c) (3x / (x-1) × 1 / (x+1) ÷ 3 / (x²-1)= 3x / (x - 1) (x + 1) ÷ 3 / (x - 1) (x + 1)
= 3x / (x - 1) (x + 1) x (x - 1) (x + 1) / 3
= x
(d) (x - 4) / (x+4) × (x-3) / (x+3) ÷ (x²-7x+12) / (x²+7x+12)= (x - 4) (x - 3) / (x + 4) (x + 3) ÷ (x - 3) (x - 4) / (x + 4)(x + 3)
= (x - 4)(x - 3) / (x + 4) (x + 3) x (x + 4) (x + 3) / (x - 3) (x - 4)
= 1
(e) (a + b) / (a - b) - (a - b) / (a + b) × (a² - b²) / 4ab= (a + b) (a + b) - (a - b) (a - b) / (a - b) (a + b) × (a + b) (a - b) / 4ab
= a² + 2ab + b² - a² + 2ab - b²/ 4ab
= 4ab / 4ab = 1
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