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Learn/ 9th Mathematics/ Unit 4 /Numericals

Unit 4: Factorization and Algebraic Manipulation — Review Exercise

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

R4.2.Factorize the following expressions: (i) 4x^3 + 18x^2 - 12x (ii) x^3 + 64y^3 (iii) x^3y^3 - 8 (iv) -x^2 - 23x - 60 (v) 2x^2 + 7x + 3 (vi) x^4 + 64 (vii) x^4 + 2x^2 + 9 (viii) (x+3)(x+4)(x+5)(x+6) - 360 (ix) (x^2+6x+3)(x^2+6x-9) + 36
Result
(i) 4x3+18x2-12x = 2x(2x2+9x-6)
(ii) x3+64y3 = (x)3+(4y)3 = (x+4y)[(x)2-(x)(4y)+(4y)2]
(ii) (x+4y)(x2-4xy+16y2)
(iii) x3y3-8 = (xy)3-(2)3 = (xy-2)[(xy)2+(xy)(2)+(2)2]
(iii) (xy-2)(x2y2+2xy+4)
(iv) -x2-23x-60 = -(x2+23x+60) = -[x(x+20)+3(x+20)]
(iv) -(x+3)(x+20)
(v) 2x2+7x+3 = 2x2+6x+x+3 = 2x(x+3)+1(x+3)
(v) (2x+1)(x+3)
(vi) x4+64 = (x2)2+(8)2+2(x2)(8)-2(x2)(8) = (x2+8)2-(4x)2
(vi) (x2-4x+8)(x2+4x+8)
(vii) x4+2x2+9 = (x2)2+2x2+(3)2+2x2-2x2 = (x2+3)2-(2x)2
(vii) (x2-2x+3)(x2+2x+3)
(viii) (x+3)(x+6)(x+4)(x+5)-360 = (x2+9x+18)(x2+9x+20)-360
Working
(viii) y=x2+9x; (y+18)(y+20)-360 = y2+38y = y(y+38)
Result
(viii) x(x+9)(x2+9x+38)
Working
(ix) y=x2+6x; (y+3)(y-9)+36 = y2-6y+9 = (y-3)2
Result
(ix) (x2+6x-3)2
R4.3.Find LCM and HCF by prime factorization method: (i) 4x^3 + 12x^2, 8x^2 + 16x (ii) x^3 + 3x^2 - 4x, x^2 - x - 6 (iii) x^2 + 8x + 16, x^2 - 16 (iv) x^3 - 9x, x^2 - 4x + 3
(i) 4x3+12x2=4x2(x+3),quad 8x2+16x=8x(x+2)
Result
(i) HCF = 4x,quad LCM = 8x2(x+2)(x+3)
(ii) x3+3x2-4x=x(x-1)(x+4),quad x2-x-6=(x-3)(x+2)
(ii) HCF = 1,quad LCM = x(x-1)(x+2)(x-3)(x+4)
(iii) x2+8x+16=(x+4)2,quad x2-16=(x-4)(x+4)
(iii) HCF = x+4,quad LCM = (x-4)(x+4)2
(iv) x3-9x=x(x-3)(x+3),quad x2-4x+3=(x-3)(x-1)
(iv) HCF = x-3,quad LCM = x(x-1)(x2-9)
R4.4.Find square root by factorization and division method of the expression 16x^4 + 8x^2 + 1.
Factorization 16x4+8x2+1 = (4x2)2+2(4x2)(1)+(1)2 = (4x2+1)2
Result
Factorization result sqrt{16x4+8x2+1} = ± (4x2+1)
Division method Division of 16x4+8x2+1 gives quotient 4x2+1, remainder 0
Division result sqrt{16x4+8x2+1} = ± (4x2+1)
R4.5.Huria is analyzing the total cost of her loan, modeled by the expression C(x) = x^2 - 8x + 15, where x represents the number of years. What is the optimal repayment period for Huria's loan?
Given
Given C(x) = x2-8x+15
Factorizing C(x) = x2-5x-3x+15 = (x-5)(x-3)
Setting to zero (x-5)(x-3)=0
Result
Result x=3 or x=5 years