Ex 4.5, 1 (vi)

Transcript

Ex 4.5, 1 (vi) Simplify the following rational expressions assuming that the expressions in the denominators are not equal to zero: (vi) (𝑝^4−16)/(𝑝^2 − 4𝑝 + 4) Factorising numerator and denominator separately Factorising Numerator We have to factorise 𝑝^4−16 Now, 𝑝^4−16=(𝒑^𝟐 )^𝟐−𝟒^𝟐 =(𝑝^2+4)(𝑝^2−4) =(𝑝^2+4)(𝑝^2−2^2 ) Using 𝑎^2−𝑏^2=(𝑎+𝑏)(𝑎−𝑏) Where 𝑎 = 𝑝^2, b = 4 =(𝑝^2+4)(𝑝^2−4) =(𝑝^2+4)(𝒑^𝟐−𝟐^𝟐 ) =(𝒑^𝟐+𝟒)(𝒑−𝟐)(𝒑+𝟐) Factorising Denominator We have to factorise 𝑝^2 − 4𝑝 + 4 p2 – 4p + 4 Factorising by splitting the middle term = p2 – 2p – 2p + 4 Using 𝑎^2−𝑏^2=(𝑎+𝑏)(𝑎−𝑏) Where 𝑎 = 𝑝, b = 2 Splitting the middle term method We need to find two numbers whose Sum = –4 Product = 1 × 4 = 4 Since sum is negative but product is positive. Thus, both numbers are negative = p(p – 2) – 2(p – 2) = (p – 2) (p – 2) Thus, our rational expression becomes (𝑝^4 − 16)/(𝑝^2 − 4𝑝 + 4)=((𝑝^2 + 4)(𝑝 − 2)(𝑝 + 2))/((𝑝 − 2)(𝑝 − 2)) =((𝒑^𝟐 + 𝟒) (𝒑 + 𝟐))/((𝒑 − 𝟐) )