Greatest Common Divisor(G.C.D) Or H.C.F
To find H.C.F follow , following Steps:
A) Obtain the polynomials. Lat the polynomials be p(x) and q(x).
B) Factorise the polynomials p(x) and q(x).
C) Express p(x) and q(x) as a product of powers of irreducible factors. Also, express the numerical factor as a product of powers of primes.
D) Identify common irreducible divisors and find the smallest (least) exponents of these common divisors in the given polynomials.
E) Raise the common irreducible divisors to the smallest exponents found in step D and multiply them to get the GCD(HCF). If there is no common divisors, the GCD (HCF) is 1.
The following formulae will be helpful in factorising polynomials
1) (x+ y)²= x² + 2xy + y²
2) (x- y)²= x² - 2xy + y²
3) (x+ y+ z)²= x² + y²+ z²+ 2xy+ 2yz+ 2zx.
4) (x+ y)³ = x³ + 3xy(x+ y) + y³
5) (x- y)³ = x³ - 3xy(x- y) - y³
6) x²- y² = (x+ y)(x-y)
7) x³+ y³ = (x+ y)(x² - xy +y²)
8) x³- y³ = (x- y)(x² + xy +y²)
9) x³+ y³ + z³ - 3xyz= (x+ y+ z)(x² +y²+ z²- xy - yz - zx)
10) x⁴- y⁴ = (x²+ y²)(x² - y²)
= (x+ y)(x - y)(x² +y²)
11) x⁸- y⁸ = (x⁴- y⁴)(x⁴ +y⁴)
= (x+ y)(x- y)(x² +y²)(x⁴ + y⁴)
12) x⁴+ y⁴+ x²y² = (x⁴+ y⁴+ 2x²y²) - x²y²
= (x² +y²)² -(xy)²
= (x²++ xy y²)(x² - xy +y²)
1) (x -2)²(x-3)²(x+1)² and (x +1)³(x+2)³(x- 3). (x+2)²(x- 3)(x+1)²
2) 30(x²- 3x+2)(x -2) & 50(x²- 2x+1)(x²- 5x+6). 10(x- 1)(x -2)
3) (x -1)²(x+ 2)(x+3)³ and (x -1)(x-2)³(x+ 3)². (x- 1)(x +3)²
4) (2x² -18) and (x² - 2x -3). x - 3
5) 2x² - x -1 and 4x² + 8x +3. 2x+1
6) ⁸(x³ -x²+ x) and 28(x³ +1). 4(x- 2).
7) 24(6x⁴- x³ -2x²) and 20(2x⁶+ 3x⁵+ x⁴). 4x²(2x+1)
8) x² + x-2 and x³ + 4x²+ x - 6. (x-1)(x +2)
9) 4x³ -16x² + 20x-8 and 8x³ - 48x²+ 72x - 32. 4(x -1)²
10) 33(2x+3)(3x -4)³(5- 4x)⁴ and 22(x +1) (2x +3)(5- 4x)³(4x- 9). 11(2x+3)²(4x -5)³
11) 10(x+1)²(x -3)³(x-2) and 15(x -2)³(x -3)²(x+5) and 75(x+5)(x-3)³(x-2)². 5(x -2)(x -3)²
12) 8x⁴ - 16x³ - 40x² + 48x and 16x⁵ + 64x⁴ + 80x³ + 32x². 8x(x +2)
13) 2x⁴ - 2y⁴ and 3x³ + 6x²y - 3xy² - 6y³. (x - y)(x + y).
14) 4(x - 3)²(x -1)(x+1)³ and 6(x- 1)²(x +1)²(x+7). 2(x-1)(x+1)²
15) 16 - 4x² and x² + x- 6. x-2
16) xy - y and x⁴y - xy. (x-1)y
17) x³ + 64 and x² - 16. x+4
18) 3 + 13x - 30x² and 25x² - 30x +9. 5x -3
19) 56(x⁶y² - x²y⁶) and 72(x⁵y³+ 3y⁵x³ + 2y⁷x). 8xy²(x²+ y²)
20) (x - 2)²(x +3)(x-4) and (x- 2)(x +2)(x -5). (x-2)
21) (2x - 7)(3x +4) and (2x- 7)²(x +3). (2x -7)
22) (3x - 2)²(2x+3)³(x -1) and (3x- 2)³(2x +3)(x-1)³. (2x+3)(x-1)(3x-2)²
23) (x +1)³(x -1) and (x- 1)³(x +1). (x-1)(x+1)
24) (x +4)²(x -3)³ and (x- 1)(x -3)²(x+4). (x+4)(x-3)²
25) 24(x - 3)(x -2)² and 15(x- 2)(x -3)³. 3(x-3)(x-2)
26) 2x² - 7x +3 and 3x²-7x - 6. (x-3)
27) x³+ 2x²- 3x and 2x³ + 5x² -3x. x(x +3)
28) 22x(x+1)² and 36x²(2x²+ 3x+1). 2x(x +1). 2x(x+1)
29) 3+ 13x - 50x² and 25x²-30x +9. (5x-3)
30) x³ - y³ and x⁴+ x²y² + y⁴. (x² + xy+ y²)
31) 2x² + 7xy + 3y² and 2x²+ 6xy + x + 3y. x + 3y
32) 56(x⁶y² - x²y⁶) and 72(x⁵y³+ 3y⁵x³+ 2y⁷x). 8xy²(x²+ y²)
33) 54(x³+ 8y³) and 90(x³ + 7x²y+ 16xy²+ 12y³). 18(x+2y)
34) (4x⁴+ y⁴) and (2x³ - xy²+ y³). 2x² + 2xy + y²
35) x³ - y³ , x³y - y⁴, y²(x - y)²(x² + xy + y²). x³ - y³
36) (2x² - 3xy)², (4x - 6y)³ and (8x³ - 27y³). 2x - 3y
37) (4x⁴ + y⁴), (2x³ - xy² - y³) and (2x² + 2xy + y²). 2x² + 2xy + y²
38) x⁴ - x³ + x -1 and x⁴+ x² +1. x² - x +1
39) (8x⁶ - 32x⁵+ 128x³ - 128x²) and (12x⁶ - 36x⁵ + 48x³). 4x²(x -2)
40) (8x³ - x²- x +1) and (x⁴ - 2x³ + 2x -1). (x- 1)²(x+1)
41) 2x²y(x²- y²) and 35xy²(x - y). xy(x- y)
42) (x² + 4x -21) and (x³+ 7x² - 9x -63). (x+7)(x-3)
43) (6x⁴- 13x³ +6x²) and (8x⁴- 3x³+ 54x² - 27). x(2x-3)
44) (4x⁵+ 16x⁴ - 44x² - 24x) and (2x⁵ - x³ - 2x). 2x(x² - x -1)
45) (3x³ - 14x² + 9x+ 10) and 15(x³-34x²+ 21x-10). 3x -5
46) 4x²(x² - a²) and 9x²(x³ - a³). x²(x - a)
47) 2(a² - b²) and 3(a³ - b³). a-- b
48) 45(2x⁴ - x³ - x²) and 75(8x⁵ + x²).
49) (x²+3x-4) and (x³- 2x²- 2x+3). x-1
50)12(x⁴ - 25) and 8(x⁴+4x²- 5). 4(x²+5)
) If x- a is the HCF of x² - x-6 and x²+ 3x - 18 find the value of a. 3
) For what value of a of x² - 2x -24 and x²- ax - 6 is x-6. 5
) x² + x- 2 is the HCF of (x -1)(2x²+ ax +2) and (x+2)(3x²+ bx +1). Find the value of a and b. 5, -4
) If x- k is the HCF of x² + x- 12 and 2x²- kx - 9. find the value of k. 3
) Find the value of a and b so that the polynomials p(x) and q(x) have (x-1)(x +4) as their HCF. 12,4
) If x- a is the HCF of ax² + bx + c and cx²+ ax + b such that c ≠ 0, then show that a³+ b³ + c³ = 3abc.
) If x- a is the HCF of x² + ax + b= 0 and x²+ bx + a= 0, then find the value of a+ b+ 1.
) If x- a is the HCF of x² + mx +1 and x²- 3x +2 find the value of m. -5/2, -2
EXERCISE
Find the LCM of following:
1) 12(x- 1)³(x³+ 6x + 8) and 150(x- 1)(x +2)(x² + 7x +10). 300(x-1)³ (x+2)²(x+4)(x+5)
2) 18x⁴ - 36x³+ 18x² and 45x⁶ - 45x³. 90x³(x -1)²(x²+ x+1)
3) 15(4x³-4x²+x) and 352x²- 7x+3). 105x(2x -1)²(x-3)
4) x³+ x²+x+1 and x³ +2x²+ x+2 (x+1)(x +2)(x²+1)
5) 7x³+ 2x²- 16x- 31 and x³+6x²+11x+ 6. (x+1)(x +2)(x-2)(x+3)(7x²+ 16x+16)
6) 3(x⁴- x²y²), 6xy(x²- y²) and 96(x³y + y⁴). 90x²y(x - y)(x+ y)(x²- xy+ y²)
7) a² - b² - c²+ 2bc, (a+ b+ c)², a² - b² + c² + 2ac. (a+ b+ c)(A-- b+ c)(a+ b- c)²
8) - x² - x+6 and -x²+ x +2. (x-2)(x+1)(x+3)
9) x³+ x²+ x+1, x³+2x² +x+2. -(x+3)²(x-5)²
10) 13(x -1)(x-2)², 7(x-2)²(x+3)² and (x-1)²(x+3). 91(x -1)²(x-2)²(x+3)²
11) 8(1- x)³+(x+2)(x-3) and 12(x+2)² (x+1) (x+3).
12) 12(x⁴+324x), 36x³+ 90x²- 54x. 36x(2x-1)(x+3)(x²- 3x+9)
13) 18x³+45x²-27x, 15x⁴ - 135x². 45x²(x-3)(x+3)(2x-1)
14) 20(2x³+3x² - 2x), 45(x⁴+ 8x). 180(x+2)(2x-1)(x²- 2x+4)
15) 35(x⁴- 27x), 40(2x³ - 5x² -3x). 280x(x-3)(2x-1)(x² + 3x+9)
16) 22x(x+1)², 36x²(2x²+ 3x+1). 396x² (x+1)²(2x+1)
17) x³ - 1, x⁴+x²+ 1, x⁶ - 1. (x³ - 1)(x³+ 1)
18) x³ - 8, x²-5x+ 6, x³ - 4x² + 4x. x(x - 2)²(x-3)(x² + 2x+4)
19) x² - 3x+2, (x-1)², (x⁴ - 1). (x+1)(x-1)²(x-2)(x²+1)
20) 15x³ - 75x²- 90x, 6x⁴ - 18x³ - 108x². 30x²(x-6)(x+1)(x+3)
21) a³+ 2a²-5a-6, a³- 3a²+4. (a+1)(a+3)(A-- 2)²
22) a² - 8a + 7, a² - 12a + 35, a³ - 5a² - a +5. (a-1)(a+1)(a-5)(a-7)
23) x² + 2xy + y², xy²+ x²y. xy(x - y)²
24) (x³- y³), x² - y². (x² - y²)(x² +xy+ y²)
25) x² + 2xy + y², x³+ y³, xy²+ x²y. xy(x + y)²(x²- xy+ y²)
26) 6x² - 13xa + 6a², 6x²+11xa-10a², 6x²+ 2ax - 4a². 2(x+a)(3x - 2a)(2x+5a)(2x - 3a).
27) x² + 2xy + y²- z², y²+z ²- x²+ 2yz, z²+ x² - y²+ 2zx. (x+ y+z)(x+y-z)(y+ z- x)(z+ x - y)
28) 6a³b²- 6ab⁴, 8a³b + 16a²b²+ 8ab³, 4a³- 2a²b - 6ab². 24ab²(a+b)(a-b)( 2a- 3b)
29) 4(a⁴ - b⁴), 2(a⁴ + b⁴ - 2a²b²), 6(a⁴ - 3a²b² - 4b²). 12(a + b)²(a- b)²(a² +b²)(a+ 2b)(a- 2b).
30) 11x³(x+ 1)³, 121x(2x² + 3x+1). 121x³(x+ 1)³(2x+1).
31) 25(x²+7x+ 12), 15x(x² - 16). 75x (x+ 3)(x+4)(x-4).
32) x(8x³- 27), 2x²(2x² +9x+9). 2x²(8x³+27)(x+3).
33) 6(x +2)²(x²-x+1) and 15(x+1)(x+2)². 30(x+1)(x+2)⁴(x²-x+1)
34) (x -3)²(x+2), (x-3)⁴, (x+2)³, (x+1)(x-3). (x - 3)⁴(x+2)³(x+1)
35) (1- x²), (x³- 1) and (x⁴-1). (x⁴-2)(x²+x+1)
Misc-
1) The LCM of two polynomial (x+ 3)(- 6x² + 5x+ 4) and (2x²+ 7x +3)(x+3) is (x+3)(2x+1)(4- 3x). Find their HCF.
2) The HCF and LCM of two polynomial are (x+ a) and 12x²(x+ a)(x² - a²) respectively. The first polynomial is 4x(x+a)². Find the other polynomials.
3) The HCF of two polynomials is (x -3)(x² +x -2) and (x²- 5x +6) is (x -3). Find their HCF.
4) The LCM of two polynomial 10 (x² - 9)(x - 1) and 10x(x - 1)(x+3) (x - 2) is 20x(x² - 9)(x² - 3x+2). Find their HCF.
5) Find the LCM and HCF of the polynomial (2x² -8) and (x² - 5x+ 6). Verify that the product of the two polynomials is equal to the product of their HCF and LCM.
6) The HCF of two polynomials is (x+ 1) and their LCM is (x⁶ -1). If one of them is (x³+1), find the other.
7) The LCM of two polynomials is (x +3)(x -2)²(x -6) and their HCF is (x -2). If one of them is (x+3)(x -2)², find the other.
8) Find GCD (x⁶y² - x²y⁶) and 72(x⁵ + 3y⁵x³+ 2y⁷x)
9) Find the LCM of (4a⁴ - b⁴), 2(a⁴ + b⁴ - 2a²b²) and 6(a⁴ - 3a²b² - 4b²).
10) The HCF and LCM of two polynomials are 5(x+3)(x -1) and 20(x² -9)(x² - 3x+2) respectively. If one polynomial is 10(x² -9)(x-1), Find thei other polynomial.