Let's take two expressions xy and yz. Here, xy is the product of x and y and yz is the product of y and z. x and y are factors of xy and y and z are the factors of yz. y is factor of both the expressions. So, y is called the highest common factor (HCF) of the expressions xy and yz.
\( \boxed {Note: HCF\:\text {divides each of the given expression exactly} } \)
Note: If there is no any common factor in the given expression, HCF = 1 as 1 is the factor of any number. |
Let's take two expressions a^{2}b - ab^{2}and a^{3}b - ab^{3}.Factorizing the expressions,
Here,
\begin{align*} first \: expression &= a^2 b - ab^2 \\ &= ab (a -b) \end{align*}
\begin{align*} second \: expression &= a^3 b - ab^3 \\ &=ab (a^2 - b^2) \\ &= ab(a + b) \: (a - b)\\ \end{align*}
common factors = ab(a - b)
Remaining factors = (a + b)
\begin{align*} LCM &= Common \: factors \times Remaining \: factor \\ &= ab(a - b) \times (a + b)\\ &= ab(a^2 - b^2).\end{align*}
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x^{3}- x^{2}-x+1,x^{4}-2x^{3}+2x-1
X^{2}+3x-4,x^{3}-2x^{2}-x+2
(a+3)^{2}-9a-27,a^{5}-13a^{3}+36a
a^{2}-ab-2b^{2},a^{3}-a^{2}b-4ab^{2}+4b^{3}
x^{2}+5x+6,x^{2}+3x+2,x^{2}-4
x^{2}+7x+12,x^{2}+4x+3,x^{2}-9
x-3
x-1
x+3
x+1
3a^{2}-8a+4,2a^{2}-5a+2,a^{4}-8a
a+1
a+2
a-2
a-1
2a^{2}-5a+2,3a^{2}-8a+4,a^{4}-8a
m^{2}-7m+12, m^{3}--2m^{2}-2m-3
(m+3)(m-4)(m^{2}-m+1)
(m-3)(m-4)(m^{2}+m+1)
(m+3)(m-4)(m^{2}+m-1)
(m+3)(m-4)(m^{2}+m+1)
m^{2}+3m-4,m^{3}-2m^{2}-2m+3
t^{2}+5t+6,t^{2}-4,t^{2}+t-6
x^{3}+5x^{2}+6x,2x^{2}+14x+24,x^{2}+6x+8
The product of two expressions is (a+1)^{3}(a-1) and LCM is (a+1)^{2}(a-1).Find their HCF.
The product of two expressions is (a+1)^{3}(a-1) and LCM is (a+1)^{2}(a-1).Find their HCF.
The product of two expressions is a(b+c)^{2}(b-c) and HCF is (b+c).Find the LCM.
The HCF and LCM of two expressions are (a-b)and b(a-b)^{2}.Find the product of the expressions.
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