Question 1 - Think & Reflect (Page 87) - Simplifying Rational Expressions - Chapter 4 Class 9 - Exploring Algebraic Identities (Ganita Manjari I)

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Question 1 - Think & Reflect (Page 87) Try to simplify the following rational expression: (36𝑠^2 − 12𝑠𝑡 + 𝑡^2)/(𝑡^2 + 2𝑡𝑠 − 48𝑠^2 )=((6𝑠−𝑡)^2)/((−+…)(−+−)). (Hint: Factor 𝑡^2+2𝑡𝑠−48𝑠^2 and simplify the rational expressions assuming that 𝑡^2+2𝑡𝑠−48𝑠^2≠0 ). Here, both numerator and denominator can be factorized using the (𝐚−𝐛)^𝟐 formula Since numerator has been factorised, let’s factorise the denominator Factorising Denominator We have to factorise 𝒕^𝟐+𝟐𝐭𝐬−𝟒𝟖𝐬^𝟐 Here, 𝒕^𝟐 and −𝟒𝟖𝒔^𝟐 are the square terms We cannot use (𝐚−𝐛)^𝟐 or (𝐚+𝒃)^𝟐 because in both these formulas, the square terms are positive – but we have negative square term −𝟒𝟖𝒔^𝟐 We do this using splitting the middle term Now, 𝑡^2+2ts−48s^2 Taking t as main variable, and s as constant Factorising by Splitting the middle term Here, 𝒕^𝟐 and −𝟒𝟖𝒔^𝟐 are the square terms We cannot use (𝐚−𝐛)^𝟐 or (𝐚+𝒃)^𝟐 because in both these formulas, the square terms are positive – but we have negative square term −𝟒𝟖𝒔^𝟐 We do this using splitting the middle term Now, 𝑡^2+2𝑠𝑡−48s^2 Taking t as main variable, and s as constant Factorising by Splitting the middle term = 𝑡^2+𝟖𝒔𝐭−𝟔𝐬𝐭−48s^2 = 𝑡(𝑡+8𝑠)−6𝑠(𝑡+8𝑠) Splitting the middle term method We need to find two numbers whose Sum = 2s Product = 1 × –48s2 = –48s2 Since product is negative, one number is negative, one is positive. And, sum is positive: so it means bigger number is positive = (𝒕+𝟖𝒔)(𝒕−𝟔𝒔) Thus, our rational expression becomes (𝟑𝟔𝒔^𝟐 − 𝟏𝟐𝒔𝒕 + 𝒕^𝟐)/(𝒕^𝟐 + 𝟐𝒕𝒔 − 𝟒𝟖𝒔^𝟐 )=(6𝑠 − 𝑡)^2/((𝑡 + 8𝑠)(𝑡 − 6𝑠)) =((6𝑠 − 𝑡)\ × (6𝑠 − 𝑡))/((𝑡 + 8𝑠)(𝑡 − 6𝑠)) Writing (𝒕 − 𝟔𝒔)=−(𝟔𝒔−𝒕) =((6𝑠 − 𝑡)\ × (6𝑠 − 𝑡))/((𝑡 + 8𝑠) × −(6𝑠 − 𝑡)) Putting negative sign on numerator =(−(6𝑠 − 𝑡)\ × (6𝑠 − 𝑡))/((𝑡 + 8𝑠) × (6𝑠 − 𝑡)) =(−(𝟔𝒔 − 𝒕)\ )/((𝒕 + 𝟖𝒔) ) =(−(6𝑠 − 𝑡)\ )/((𝑡 + 8𝑠) ) =(𝑡 − 6𝑠)/((𝑡 + 8𝑠) ) =(𝒕 − 𝟔𝒔)/(𝒕 + 𝟖𝒔)