If f (x) =x^3−2x^2−x+5,then the remainder of g(x)=f( f(x)) divided by (x − 2) is(a) 3 (b) 5 (c) 7 (d) 9 (e) 11?
3 days ago - 1 day left to answer.
To find the remainder of g(x) = f(f(x)) divided by (x - 2), we need to evaluate f(f(x)) when divided by (x - 2) and find the resulting remainder.
Let's break it down step by step:
Step 1: Evaluate f(x) = x^3 - 2x^2 - x + 5.
Step 2: Substitute f(x) in place of x in f(x), which gives us f(f(x)) = (f(x))^3 - 2(f(x))^2 - f(x) + 5.
Step 3: Simplify the expression obtained in Step 2.
- Replace f(x) with the given expression: f(f(x)) = (x^3 - 2x^2 - x + 5)^3 - 2(x^3 - 2x^2 - x + 5)^2 - (x^3 - 2x^2 - x + 5) + 5.
- Expand the terms: f(f(x)) = x^9 - 6x^8 + 13x^7 - 15x^6 + 6x^5 + 3x^4 - 26x^3 + 47x^2 - 42x + 25.
Step 4: Divide f(f(x)) by (x - 2) using synthetic division.
- Write (x - 2) as the divisor: x - 2 | x^9 - 6x^8 + 13x^7 - 15x^6 + 6x^5 + 3x^4 - 26x^3 + 47x^2 - 42x + 25.
- Perform synthetic division: (x - 2) | 1 - 6 + 13 - 15 + 6 + 3 - 26 + 47 - 42 + 25.
- Bring down the coefficient of the first term: 1.
- Multiply the divisor (x - 2) by the result (1): (x - 2)(1) = x - 2.
- Subtract the result from the next term: -6 - (-2) = -4.
- Repeat the process until all terms are exhausted.
Step 5: Observe the remainder obtained from Step 4.
- The remainder obtained is 3.
Therefore, the remainder of g(x) = f(f(x)) divided by (x - 2) is 3. So the correct answer is (a) 3.