Ex 4.5, 1 (v)

Transcript

Ex 4.5, 1 (v) Simplify the following rational expressions assuming that the expressions in the denominators are not equal to zero: (v) (𝑥^2 + 𝑥 − 6)(𝑥^2 − 7𝑥 + 12)/(𝑥^2 − 6𝑥 + 8)(𝑥^2 − 9) We have 4 terms here, and we will factorise them separately Factorising (𝒙^𝟐 + 𝒙 − 𝟔) x2 + x – 6 Factorising by splitting the middle term = x2 + 3x – 2x – 6 = x(x + 3) – 2(x + 3) = (x – 2) (x + 3) Splitting the middle term method We need to find two numbers whose Sum = 1 Product = 1 × –6 = –6 Since product is negative, one number is negative, one is positive. And, sum is positive: so it means bigger number is positive Factorsing (𝒙^𝟐 − 𝟕𝒙 + 𝟏𝟐) x2 – 7x + 12 Factorising by splitting the middle term = x2 – 4x – 3x + 12 = x(x – 4) – 3(x – 4) = (x – 4) (x – 3) Splitting the middle term method We need to find two numbers whose Sum = –7 Product = 1 × 12 = 12 Since sum is negative but product is positive. Thus, both numbers are negative Factorsing (𝒙^𝟐 −𝟔𝒙 +𝟖) x2 – 6x + 8 Factorising by splitting the middle term = x2 – 4x – 2x + 8 = x(x – 4) – 2(x – 4) = (x – 4) (x – 2) Factorsing (𝒙^𝟐 −𝟗) x2 – 9 = x2 – 32 Using 𝑎^2−𝑏^2=(𝑎+𝑏)(𝑎−𝑏) Where 𝑎 = 𝑥, b = 3 = (x – 3) (x + 3) Thus, our rational expression becomes (𝒙^𝟐 + 𝒙 − 𝟔)(𝒙^𝟐 − 𝟕𝒙 + 𝟏𝟐)/(𝒙^𝟐 − 𝟔𝒙 + 𝟖)(𝒙^𝟐 − 𝟗) = ((𝒙 − 𝟐)(𝒙 + 𝟑)(𝒙 − 𝟒)(𝒙 − 𝟑))/((𝒙 − 𝟒)(𝒙 − 𝟐)(𝒙 + 𝟑)(𝒙 − 𝟑)) = 𝟏