Ex 4.5, 1 (iv)

Transcript

Ex 4.5, 1 (iv) Simplify the following rational expressions assuming that the expressions in the denominators are not equal to zero: (iv) (4𝑦^2 − 20𝑦𝑧 + 25𝑧^2)/((25𝑧^2 − 4𝑦^2 ) ) Factorising numerator and denominator separately Factorising Numerator We have to factorise 4𝑦^2−20𝑦𝑧 +25𝑧^2 This looks like (𝒂−𝒃)^𝟐 formula Now, 𝟒𝒚^𝟐−𝟐𝟎𝒚𝒛 +𝟐𝟓𝒛^𝟐 =4𝑦^2+25𝑧^2−20𝑦𝑧 =(𝟐𝐲)^𝟐+(𝟓𝒛)^𝟐−𝟐 × 𝟐𝒚 × 𝟓𝒛 =(𝟐𝒚−𝟓𝒛)^𝟐 Factorising Denominator We have to factorise 25𝑧^2 − 4𝑦^2 This looks like 𝒂^𝟐−𝒃^𝟐 formula Now, 25𝑧^2 − 4𝑦^2 =(𝟓𝒛)^𝟐−(𝟐𝒚)^𝟐 Using (𝑎−𝑏)^2 = 𝑎^2 + 𝑏^2 – 2ab Where 𝑎 = 2𝑦, b = 5𝑧 Using 𝑎^2−𝑏^2=(𝑎+𝑏)(𝑎−𝑏) Where 𝑎 = 5𝑧, b = 2𝑦 =(𝟓𝒛−𝟐𝒚)(𝟓𝒛+𝟐𝒚) Thus, our rational expression becomes (4𝑦^2 − 20𝑦𝑧 + 25𝑧^2)/((25𝑧^2 − 4𝑦^2 ) )=(𝟐𝒚 − 𝟓𝒛)^𝟐/(𝟓𝒛 − 𝟐𝒚)(𝟓𝒛 + 𝟐𝒚) Now, we can write (𝟐𝒚 −𝟓𝒛)^𝟐=[−(5𝑧−2𝑦)]^2 =[−1 × (5𝑧−2𝑦)]^2 =(−1)^2 × (5𝑧−2𝑦)^2 = (𝟓𝒛−𝟐𝒚)^𝟐 So, our expression becomes (4𝑦^2 − 20𝑦𝑧 + 25𝑧^2)/((25𝑧^2 − 4𝑦^2 ) )=(𝟓𝒛 − 𝟐𝒚)^𝟐/(𝟓𝒛 − 𝟐𝒚)(𝟓𝒛 + 𝟐𝒚) =((5𝑧 − 2𝑦) ×(5𝑧 − 2𝑦))/(5𝑧 − 2𝑦)(5𝑧 + 2𝑦) =((𝟓𝒛−𝟐𝒚))/((𝟓𝒛+𝟐𝒚))