StudentAidUnit 4: Factorization and Algebraic Manipulation — Exercise 4 2
9th Class Mathematics · Unit 4: Factorization and Algebraic Manipulation
4.2.1.Factorize each of the following expressions: (i) 4x^4 + 81y^4 (ii) a^4 + 64b^4 (iii) x^4 + 4x^2 + 16 (iv) x^4 - 14x^2 + 1 (v) x^4 - 30x^2y^2 + 9y^4 (vi) x^4 - 7x^2y^2 + y^4
(i)
4x4+81y4 = (2x2)2+(9y2)2+2(2x2)(9y2)-2(2x2)(9y2) = (2x2+9y2)2-(6xy)2
Result
(i)
(2x2+9y2-6xy)(2x2+9y2+6xy)
(ii)
a4+64b4 = (a2)2+(8b2)2+2(a2)(8b2)-2(a2)(8b2) = (a2+8b2)2-(4ab)2
(ii)
(a2+8b2+4ab)(a2+8b2-4ab)
(iii)
x4+4x2+16 = x4+8x2+16-4x2 = (x2+4)2-(2x)2
(iii)
(x2+4-2x)(x2+4+2x)
(iv)
x4-14x2+1 = x4+2x2+1-16x2 = (x2+1)2-(4x)2
(iv)
(x2+1-4x)(x2+1+4x)
(v)
x4-30x2y2+9y4 = x4+6x2y2+9y4-36x2y2 = (x2+3y2)2-(6xy)2
(v)
(x2-6xy+3y2)(x2+6xy+3y2)
(vi)
x4-7x2y2+y4 = x4+2x2y2+y4-9x2y2 = (x2+y2)2-(3xy)2
(vi)
(x2-3xy+y2)(x2+3xy+y2)
4.2.2.Factorize each of the following expressions: (i) (x+1)(x+2)(x+3)(x+4)+1 (ii) (x+2)(x-7)(x-4)(x-1)+17 (iii) (2x^2+7x+3)(2x^2+7x+5)+1 (iv) (3x^2+5x+3)(3x^2+5x+5)-3 (v) (x+1)(x+2)(x+3)(x+6)-3x^2 (vi) (x+1)(x-1)(x+2)(x-2)-16x^2
(i)
(x+1)(x+4)(x+2)(x+3)+1 = (x2+5x+4)(x2+5x+6)+1
Working
(i)
y=x2+5x; (y+4)(y+6)+1 = y2+10y+25 = (y+5)2
Result
(i)
(x2+5x+5)2
(ii)
(x+2)(x-4)(x-7)(x-1)+17 = (x2-5x-14)(x2-5x+4)+17
Working
(ii)
y=x2-5x; (y-14)(y+4)+17 = y2-10y-39 = (y-13)(y+3)
Result
(ii)
(x2-5x-13)(x2-5x+3)
Working
(iii)
y=2x2+7x; (y+3)(y+5)+1 = y2+8y+16 = (y+4)2
Result
(iii)
(2x2+7x+4)2
Working
(iv)
y=3x2+5x; (y+3)(y+5)-3 = y2+8y+12 = (y+6)(y+2)
Result
(iv)
(3x2+5x+6)(3x2+5x+2)
(v)
(x+1)(x+6)(x+2)(x+3)-3x2 = (x2+7x+6)(x2+5x+6)-3x2
Working
(v)
y=x2+6; (y+7x)(y+5x)-3x2 = y2+12xy+35x2-3x2 = y2+12xy+32x2
(v)
y2+8xy+4xy+32x2 = y(y+8x)+4x(y+8x) = (y+8x)(y+4x)
Result
(v)
(x2+8x+6)(x2+4x+6)
(vi)
(x+1)(x+2)(x-1)(x-2)-16x2 = (x2+3x+2)(x2-3x+2)-16x2
Working
(vi)
y=x2+2; (y+3x)(y-3x)-16x2 = y2-9x2-16x2 = y2-25x2 = (y-5x)(y+5x)
Result
(vi)
(x2-5x+2)(x2+5x+2)
4.2.3.Factorize: (i) 8x^3 + 12x^2 + 6x + 1 (ii) 27a^3 + 108a^2b + 144ab^2 + 64b^3 (iii) x^3 + 18x^2y + 108xy^2 + 216y^3 (iv) 8x^3 - 125y^3 + 150xy^2 - 60x^2y
Formula
(a+b)3 = a3+3a2b+3ab2+b3,quad (a-b)3 = a3-3a2b+3ab2-b3
(i)
8x3+12x2+6x+1 = (2x)3+3(2x)2(1)+3(2x)(1)2+(1)3
Result
(i)
(2x+1)3
(ii)
27a3+108a2b+144ab2+64b3 = (3a)3+3(3a)2(4b)+3(3a)(4b)2+(4b)3
(ii)
(3a+4b)3
(iii)
x3+18x2y+108xy2+216y3 = (x)3+3(x)2(6y)+3(x)(6y)2+(6y)3
(iii)
(x+6y)3
(iv)
8x3-125y3+150xy2-60x2y = (2x)3+(-5y)3+3(2x)(-5y)2+3(2x)2(-5y)
(iv)
(2x-5y)3
4.2.4.Factorize: (i) 125a^3 - 1 (ii) 64x^3 + 125 (iii) x^6 - 27 (iv) 1000a^3 + 1 (v) 343x^3 + 216 (vi) 27 - 512y^3
Formula
a3+b3=(a+b)(a2-ab+b2),quad a3-b3=(a-b)(a2+ab+b2)
(i)
125a3-1 = (5a)3-(1)3 = (5a-1)[(5a)2+(5a)(1)+(1)2]
Result
(i)
(5a-1)(25a2+5a+1)
(ii)
64x3+125 = (4x)3+(5)3 = (4x+5)[(4x)2-(4x)(5)+(5)2]
(ii)
(4x+5)(16x2-20x+25)
(iii)
x6-27 = (x2)3-(3)3 = (x2-3)[(x2)2+(x2)(3)+(3)2]
(iii)
(x2-3)(x4+3x2+9)
(iv)
1000a3+1 = (10a)3+(1)3 = (10a+1)[(10a)2-(10a)(1)+(1)2]
(iv)
(10a+1)(100a2-10a+1)
(v)
343x3+216 = (7x)3+(6)3 = (7x+6)[(7x)2-(7x)(6)+(6)2]
(v)
(7x+6)(49x2-42x+36)
(vi)
27-512y3 = (3)3-(8y)3 = (3-8y)[(3)2+(3)(8y)+(8y)2]
(vi)
(3-8y)(9+24y+64y2)