Question 29 of 43advanced🔧 ApplyFill in the Blanks3 marks

Try to simplify the following rational expression: (36s236s^{2} - 12st + t2t^{2}) / (t2t^{2} + 2ts - 48s248s^{2}) = (6s - t)^2 / (___ + )( + ___) (Hint: Factor t2t^{2} + 2ts - 48s248s^{2} and simplify the rational expressions assuming that t2t^{2} + 2ts - 48s248s^{2} ≠\neq 0).

Correct Answer

t, -6s, t, 8s

Exercise: Think and Reflect | Q: 1 | (Page 20)
For More Understanding

Explanation

To factor the denominator t2+2ts−48s2t^2 + 2ts - 48s^2, we need to find two terms whose sum is 2ts2ts and whose product is −48s2-48s^2. These two terms are 8ts8ts and −6ts-6ts. By splitting the middle term and grouping, we get t2+8ts−6ts−48s2=t(t+8s)−6s(t+8s)=(t−6s)(t+8s)t^2 + 8ts - 6ts - 48s^2 = t(t + 8s) - 6s(t + 8s) = (t - 6s)(t + 8s). Fitting this into the given format (___+___)(___+___)(\_\_\_ + \_\_\_)(\_\_\_ + \_\_\_), we get (t+−6s)(t+8s)(t + -6s)(t + 8s).

Solution Steps

  1. Step 1: Consider the denominator t2+2ts−48s2t^2 + 2ts - 48s^2.

  2. Step 2: Find two numbers that multiply to −48-48 and add to 22. These are 88 and −6-6.

  3. Step 3: Split the middle term: t2+8ts−6ts−48s2t^2 + 8ts - 6ts - 48s^2.

  4. Step 4: Factor by grouping: t(t+8s)−6s(t+8s)=(t−6s)(t+8s)t(t + 8s) - 6s(t + 8s) = (t - 6s)(t + 8s).

  5. Step 5: Express (t−6s)(t - 6s) as (t+−6s)(t + -6s) to match the given blank format (___+___)(___+___)(\_\_\_ + \_\_\_)(\_\_\_ + \_\_\_), resulting in (t+−6s)(t+8s)(t + -6s)(t + 8s).

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