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secondary 2 | Maths
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How to factorise?
Date Posted: 3 years ago
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(x + y + z)(yz + zx + xy) - xyz
= (x + y)(yz + zx + xy) + z(yz + zx + xy) - xyz
= (x + y)(yz + zx + xy) + z(yz + zx) + z(xy) - xyz
= (x + y)(yz + zx + xy) + z²(x + y) + xyz - xyz
= (x + y)(yz + zx + xy) + z²(x + y)
= (x + y)(yz + zx + xy + z²)
= (x + y)(xy + yz + xz + z²)
= (x + y)(y(x + z) + z(x + z))
= (x + y)(x + z)(y + z)
a²b - a²c - ac² - ab² - b²c + bc² + 2abc
= a²b - ab² + abc - b²c - a²c - ac² + bc² + abc
= ab(a - b) + bc(a - b) - ac(a + c) + bc(a + c)
= (a - b)(ab + bc) + (a + c)(bc - ac)
= (a - b)(a + c)(b) + (a + c)(b - a)(c)
= (a - b)(a + c)(b) - (a + c)(a - b)(c)
= (a - b)(a + c)(b - c)
= (x + y)(yz + zx + xy) + z(yz + zx + xy) - xyz
= (x + y)(yz + zx + xy) + z(yz + zx) + z(xy) - xyz
= (x + y)(yz + zx + xy) + z²(x + y) + xyz - xyz
= (x + y)(yz + zx + xy) + z²(x + y)
= (x + y)(yz + zx + xy + z²)
= (x + y)(xy + yz + xz + z²)
= (x + y)(y(x + z) + z(x + z))
= (x + y)(x + z)(y + z)
a²b - a²c - ac² - ab² - b²c + bc² + 2abc
= a²b - ab² + abc - b²c - a²c - ac² + bc² + abc
= ab(a - b) + bc(a - b) - ac(a + c) + bc(a + c)
= (a - b)(ab + bc) + (a + c)(bc - ac)
= (a - b)(a + c)(b) + (a + c)(b - a)(c)
= (a - b)(a + c)(b) - (a + c)(a - b)(c)
= (a - b)(a + c)(b - c)
Alternative for the second one :
a²b - a²c - ac² - ab² - b²c + bc² + 2abc
= a²b + abc - a²c - ac² + abc - ab² + bc² - b²c
= ab(a + c) - ac(a + c) + ab(c - b) + bc(c - b)
= (a + c)(ab - ac) + (c - b)(ab + bc)
= (a + c)(a)(b - c) + (c - b)(b)(a + c)
= (a + c)(b - c)(a) - (a + c)(b - c)(b)
= (a + c)(b - c)(a - b)
a²b - a²c - ac² - ab² - b²c + bc² + 2abc
= a²b + abc - a²c - ac² + abc - ab² + bc² - b²c
= ab(a + c) - ac(a + c) + ab(c - b) + bc(c - b)
= (a + c)(ab - ac) + (c - b)(ab + bc)
= (a + c)(a)(b - c) + (c - b)(b)(a + c)
= (a + c)(b - c)(a) - (a + c)(b - c)(b)
= (a + c)(b - c)(a - b)
Alternative for the first one :
(x + y + z)(yz + zx + xy) - xyz
= ((x + y) + z)(x(y + z) + yz) - xyz
= (x + y)(x)(y + z) + (x + y)(yz) + (xz)(y + z) + yz² - xyz
= (x + y)(y + z)(x) + (x + y)(yz) + xyz + xz(z) + yz(z) - xyz
= (x + y)(y + z)(x) + (x + y)(y)(z) + (z)(z)(x + y)
= (x + y)(y + z)(x) + (x + y)(z)(y + z)
= (x + y)(y + z)(x + z)
(x + y + z)(yz + zx + xy) - xyz
= ((x + y) + z)(x(y + z) + yz) - xyz
= (x + y)(x)(y + z) + (x + y)(yz) + (xz)(y + z) + yz² - xyz
= (x + y)(y + z)(x) + (x + y)(yz) + xyz + xz(z) + yz(z) - xyz
= (x + y)(y + z)(x) + (x + y)(y)(z) + (z)(z)(x + y)
= (x + y)(y + z)(x) + (x + y)(z)(y + z)
= (x + y)(y + z)(x + z)
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