Example 15 - Chapter 4 Class 9 - Exploring Algebraic Identities (Ganita Manjari I)

Transcript

Example 15 The sum of three numbers is 10 and their product is 25. The sum of their squares is 38 . Try to use the previous identity to find the sum of the cubes of these three numbers. Here, we have to use this identity x3 + y3 + z3 − 3xyz = (x + y + z) (x2 + y2 + z2 − xy − yz − zx) Let the 3 numbers be x, y and z Now, given that Sum of 3 numbers = 10 x + y + z = 10 And, Product of 3 numbers = 25 x × y × z = 25 xyz = 25 Also, given that Sum of their squares = 38 x2 + y2 + z2 = 38 We need to find sum of cubes of these three numbers i.e. we need to find x3 + y3 + z3 Now, putting values in our identity x3 + y3 + z3 − 3xyz = (x + y + z) (x2 + y2 + z2 − xy − yz − zx) x3 + y3 + z3 − 3 × 25 = 10 × (38 − xy − yz − zx) x3 + y3 + z3 − 75 = 10 × (38 − xy − yz − zx) x3 + y3 + z3 = 10 × (38 − xy − yz − zx) + 75 x3 + y3 + z3 = 10 × [38 − (xy + yz + zx)] + 75 We have to find xy + yz + zx To do that, we use the identity (𝒙+𝒚+𝒛)^𝟐=𝒙^𝟐+𝒚^𝟐+𝒛^𝟐+𝟐𝒙𝒚+𝟐𝒙𝒛+𝟐𝒚𝒛 Putting x + y + z = 10, x2 + y2 + z2 = 38 in our equation (𝟏𝟎)^𝟐=𝟑𝟖+𝟐𝒙𝒚+𝟐𝒙𝒛+𝟐𝒚𝒛 Product of 3 numbers = 25 x × y × z = 25 xyz = 25 Also, given that Sum of their squares = 38 x2 + y2 + z2 = 38 We need to find sum of cubes of these three numbers i.e. we need to find x3 + y3 + z3 Now, putting values in our identity x3 + y3 + z3 − 3xyz = (x + y + z) (x2 + y2 + z2 − xy − yz − zx) x3 + y3 + z3 − 3 × 25 = 10 × (38 − xy − yz − zx) x3 + y3 + z3 − 75 = 10 × (38 − xy − yz − zx) x3 + y3 + z3 = 10 × (38 − xy − yz − zx) + 75 x3 + y3 + z3 = 10 × [38 − (xy + yz + zx)] + 75 We have to find xy + yz + zx To do that, we use the identity (𝒙+𝒚+𝒛)^𝟐=𝒙^𝟐+𝒚^𝟐+𝒛^𝟐+𝟐𝒙𝒚+𝟐𝒙𝒛+𝟐𝒚𝒛 Putting x + y + z = 10, x2 + y2 + z2 = 38 in our equation (𝟏𝟎)^𝟐=𝟑𝟖+𝟐𝒙𝒚+𝟐𝒙𝒛+𝟐𝒚𝒛 100=38+2𝑥𝑦+2𝑥𝑧+2𝑦𝑧 100−38=2𝑥𝑦+2𝑥𝑧+2𝑦𝑧 62=2𝑥𝑦+2𝑥𝑧+2𝑦𝑧 2𝑥𝑦+2𝑥𝑧+2𝑦𝑧=62 2(𝑥𝑦+𝑥𝑧+𝑦𝑧)=62 𝑥𝑦+𝑥𝑧+𝑦𝑧=62/2 𝒙𝒚+𝒙𝒛+𝒚𝒛=𝟑𝟏 Putting 𝑥𝑦+𝑥𝑧+𝑦𝑧=31 in (1) x3 + y3 + z3 = 10 × [38 − (xy + yz + zx)] + 75 x3 + y3 + z3 = 10 × [38 − 31] + 75 x3 + y3 + z3 = 10 × 7 + 75 x3 + y3 + z3 = 70 + 75 x3 + y3 + z3 = 145