if a polynomial f(x) = x⁴ -2x³ + 3x² -ax-b leaves remainder 5 and 19 when divided by (x-) and (x+1) respectively . find the values of a and b . hence determine the remainder when f(x) is divided by (x-2)
Answers
Answer:
p(x) = x⁴-2x³+3x²-ax+b
The remainder of (x-1) is 6, so let’s substitute in x=1
p(1) =1⁴-2(1)³+3(1)²-a(1)+b=6
1–2+3-a+b=6
-a+b=6–2
-a+b=4 ————1 (here we have the first equation)
The remainder of (x+1) is 14, so let’s substitute in x=-1
p(-1) =(-1)⁴-2(-1)³+3(-1)²-a(-1)+b=14
1–2(-1)+3(1)+a+b=14
a+b=14–3–3
a+b=8 ————2 (here is the second equation)
Solving the 2 equations simultaneously, you will get a=2, b=6
Let’s put these values into the polynomial first.
p(x) =x⁴-2x³+3x²-2x+6
You’re looking for the remainder when the polynomial is divided by (x-2), so we’ll put in x=2
p(2) =2⁴-2(4)³+3(4)²-2(4)+6
16–2(64)+3(16)-8+6= -66
The remainder is -66
Step-by-step explanation:
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Answer:
Step-by-step explanation:
Now, the polynomial of the remainders are here given below.
