Ex 4.5, 1 (ii)

Transcript

Ex 4.5, 1 (ii) Simplify the following rational expressions assuming that the expressions in the denominators are not equal to zero: (ii) (𝑛^3 − 3𝑛^2 𝑚 + 3𝑛𝑚^2 − 𝑚^3)/(5𝑚^2 − 10𝑚𝑛 + 5𝑛^2 ) Factorising numerator and denominator separately Factorising Numerator We have to factorise 𝑛^3 − 3𝑛^2 𝑚 + 3𝑛𝑚^2 − 𝑚^3 This looks like (𝒂−𝒃)^𝟑=𝒂^𝟑−𝟑𝒂^𝟐 𝒃+𝟑𝒂𝒃^𝟐−𝒃^𝟑 Thus, 𝑛^3 − 3𝑛^2 𝑚 + 3𝑛𝑚^2 − 𝑚^3=(𝒏−𝒎)^𝟑 Factorising Denominator We have to factorise 5𝑚^2 − 10𝑚𝑛 + 5𝑛^2 Now, 5𝑚^2 − 10𝑚𝑛 + 5𝑛^2 Since 5 is multiplied to each term, we take it common =5(5/5 𝑚^2−10/5 𝑚𝑛+5/5 𝑛^2) =𝟓(𝒎^𝟐−𝟐𝒎𝒏+𝒏^𝟐) Using (𝑎−𝑏)^2 = 𝑎^2 + 𝑏^2 – 2ab Where 𝑎 = 𝑚, b = 𝑛 =𝟓(𝒎−𝒏)^𝟐 Thus, our rational expression becomes (𝑛^3 − 3𝑛^2 𝑚 + 3𝑛𝑚^2 − 𝑚^3)/(5𝑚^2 − 10𝑚𝑛 + 5𝑛^2 ) =(𝒏 − 𝒎)^𝟑/(𝟓(𝒎 − 𝒏)^𝟐 ) Now, we can write (𝒎 − 𝒏)^𝟐=[−(𝑛−𝑚)]^2 =[−1 × (𝑛−𝑚)]^2 =(−1)^2 × (𝑛−𝑚)^2 = (𝒏−𝒎)^𝟐 So, our expression becomes (𝑛^3 − 3𝑛^2 𝑚 + 3𝑛𝑚^2 − 𝑚^3)/(5𝑚^2 − 10𝑚𝑛 + 5𝑛^2 ) =(𝑛 − 𝑚)^3/(5〖(𝑛 − 𝑚)^2〗^2 ) =((𝒏 − 𝒎))/𝟓