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Learn/ 9th Mathematics/ Unit 4 /Numericals

Unit 4: Factorization and Algebraic Manipulation — Exercise 4 3

9th Class Mathematics · Unit 4: Factorization and Algebraic Manipulation

4.3.1.Find HCF by factorization method. (i) 21x^2y, 35xy^2 (ii) 4x^2 - 9y^2, 2x^2 - 3xy (iii) x^3 - 1, x^2 + x + 1 (iv) a^3 + 2a^2 - 3a, 2a^3 + 5a^2 - 3a (v) t^2 - 3t - 4, t^2 + 5t + 4, t^2 - 1 (vi) x^2 + 15x + 56, x^2 + 5x - 24, x^2 + 8x
(i) 21x2y = 3 × 7 × x × x × y,quad 35xy2 = 5 × 7 × x × y × y
Result
(i) HCF = 7xy
(ii) 4x2-9y2=(2x-3y)(2x+3y),quad 2x2-3xy=x(2x-3y)
(ii) HCF = 2x-3y
(iii) x3-1=(x-1)(x2+x+1)
(iii) HCF = x2+x+1
(iv) a3+2a2-3a=a(a+3)(a-1),quad 2a3+5a2-3a=a(a+3)(2a-1)
(iv) HCF = a(a+3)
(v) t2-3t-4=(t-4)(t+1),quad t2+5t+4=(t+4)(t+1),quad t2-1=(t-1)(t+1)
(v) HCF = t+1
(vi) x2+15x+56=(x+8)(x+7),quad x2+5x-24=(x+8)(x-3),quad x2+8x=x(x+8)
(vi) HCF = x+8
4.3.2.Find HCF of the following expressions by using division method: (i) 27x^3 + 9x^2 - 3x - 10, 3x - 2 (ii) x^3 - 9x^2 + 23x - 15, x^2 - 4x + 3 (iii) 2x^3 + 2x^2 + 2x + 2, 6x^3 + 12x^2 + 6x + 12 (iv) 2x^3 - 4x^2 - 16x, x^3 - 4x, 3x^2 + 6x
(i) (27x3+9x2-3x-10)div(3x-2) = 9x2+9x+5, remainder 0
Result
(i) HCF = 3x-2
(ii) (x3-9x2+23x-15)div(x2-4x+3) = x-5, remainder 0
(ii) HCF = x2-4x+3
(iii) (6x3+12x2+6x+12)div(2x3+2x2+2x+2) = 3, remainder 6x2+6; then (2x3+2x2+2x+2)div(6x2+6)=x3, remainder 2x2+2; (6x2+6)div(2x2+2)=3, remainder 0
(iii) HCF = 2x2+2 = 2(x2+1)
(iv) 3x2+6x=3(x2+2x); (x3-4x)div(x2+2x)=x-2, remainder 0; (2x3-4x2-16x)div(x2+2x)=2x-8, remainder 0
(iv) HCF = x2+2x = x(x+2)
4.3.3.Find LCM of the following expressions by using prime factorization method. (i) 2a^2b, 4ab^2, 6ab (ii) x^2 + x, x^3 + x^2 (iii) a^2 - 4a + 4, a^2 - 2a (iv) x^4 - 16, x^3 - 4x (v) 16 - 4x^2, x^2 + x - 6, 4 - x^2
(i) 2a2b=2 × a × a × b, 4ab2=2 × 2 × a × b × b, 6ab=2 × 3 × a × b
Result
(i) LCM = 12a2b2
(ii) x2+x=x(x+1),quad x3+x2=x2(x+1)
(ii) LCM = x2(x+1)
(iii) a2-4a+4=(a-2)2,quad a2-2a=a(a-2)
(iii) LCM = a(a-2)2
(iv) x4-16=(x-2)(x+2)(x2+4),quad x3-4x=x(x-2)(x+2)
(iv) LCM = x(x4-16)
(v) 16-4x2=4(2-x)(2+x),quad x2+x-6=(x+3)(x-2),quad 4-x2=(2-x)(2+x)
(v) LCM = 4(x2-4)(x+3)
4.3.4.The HCF of two polynomials is y - 7 and their LCM is y^3 - 10y^2 + 11y + 70. If one of the polynomials is y^2 - 5y - 14, find the other.
Given
Given HCF = y-7,quad LCM = y3-10y2+11y+70,quad p(y)=y2-5y-14
Formula
Formula p(y) × q(y) = HCF × LCM
Working
Substituting (y2-5y-14) × q(y) = (y-7)(y3-10y2+11y+70)
Simplifying q(y) = (y-7)(y3-10y2+11y+70)(y-7)(y+2) = y3-10y2+11y+70y+2
Dividing (y3-10y2+11y+70)div(y+2) = y2-12y+35, remainder 0
Result
Other polynomial q(y) = y2-12y+35
4.3.5.The LCM and HCF of two polynomial p(x) and q(x) are 36x^3(x+a)(x^3-a^3) and x^2(x-a) respectively. If p(x) = 4x^2(x^2-a^2), find q(x).
Given
Given HCF = x2(x-a),quad LCM = 36x3(x+a)(x3-a3),quad p(x) = 4x2(x2-a2)
Formula
Formula p(x) × q(x) = HCF × LCM
Working
Substituting 4x2(x2-a2) × q(x) = x2(x-a) × 36x3(x+a)(x3-a3)
Simplifying q(x) = x2(x-a) × 36x3(x+a)(x3-a3)4x2(x2-a2)
Result
Result q(x) = 9x3(x3-a3)
4.3.6.The HCF and LCM of two polynomials is (x + a) and 12x^2(x + a)(x^2 - a^2) respectively. Find the product of the two polynomials.
Given
Given HCF = x+a,quad LCM = 12x2(x+a)(x2-a2)
Formula
Formula p(x) × q(x) = HCF × LCM
Working
Substituting p(x) × q(x) = (x+a) × 12x2(x+a)(x2-a2)
Simplifying p(x) × q(x) = 12x2(x+a)2(x2-a2) = 12x2(x+a)2(x-a)(x+a)
Result
Result p(x) × q(x) = 12x2(x+a)3(x-a)