Use the remainder theorem to find the remainder whenP(x)=x³-4x²+5-2is dividid by x-22. Find the remainder whenP(x)=2x⁴+3x³-x+5is dividid x+13. Determine the remainder whenP(x)=4x³+2x²-7x+10is dividid by x-3What is the answer and solving
Answer (1)
Answer:Let's analyze each problem step-by-step using the Remainder Theorem, which states that the remainder when a polynomial P(x) is divided by (x - a) is simply P(a). 1. For P(x) = x³ - 4x² + 5 - 2, we need to find the remainder when divided by x - 2. Since the divisor is x - 2, we evaluate P(2): P(2) = (2)³ - 4(2)² + 5 - 2 = 8 - 4(4) + 5 - 2 = 8 - 16 + 5 - 2 = (8 - 16) + (5 - 2) = -8 + 3 = -5. Therefore, the remainder is -5.2. For P(x) = 2x⁴ + 3x³ - x + 5, dividing by x + 1 means we evaluate at x = -1: P(-1) = 2(-1)⁴ + 3(-1)³ - (-1) + 5 = 2(1) + 3(-1) - (-1) + 5 = 2 -3 +1 +5 = (2 -3) + (1 +5) = -1 +6 = 5. So, the remainder is 5.3. For P(x)=4x³+2x²-7x+10 divided by x-3, evaluate at x=3: P(3)=4(3)³+2(3)²-7(3)+10=4(27)+2(9)-21+10=108+18-21+10= (108+18)+(-21+10)=126-11=115. Hence, the remainder is 115.In summary:- When dividing x³ -4x² +5 -2 by x-2, the remainder is **-5**.- When dividing 2x⁴+3x³-x+5 by x+1, the remainder is **5**.- When dividing 4x³+2x²-7x+10 by x-3, the remainder is **115**.Step-by-step explanation:
