Divide 3x3 + x2 +2x +5 by 3x -1 and verify the results with division algorithm
Answers
Let f(x) = 3x² - x³ - 3x + 5 and g(x) = x - 1 - x²
Long Division Method:
-x² + x - 1) -x³ + 3x² - 3x + 5(x - 2
-x³ + x² - x
-------------------------
2x² - 2x + 5
2x² - 2x + 2
----------------------
3
Verification:
Dividend = Divisor * Quotient + Remainder
= (-x² + x - 1) - (x - 2) + 3
= -x³ + x² - x + 2x² - 2x + 2 + 3
= 3x² - x³ - 3x + 5
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Note that the given polynomials are not in standard form. To carry out division, we first write both the dividend and divisor in decreasing orders of their degrees. So, dividend = –x3 + 3x2 – 3x + 5 and divisor = –x2 + x – 1.
Division process is shown above. We stop here since deg(3) = 0 < 2 = deg(–x2 + x – 1). So, quotient = x – 2, remainder = 3.
Now,
Divisor Quotient + Remainder=
(–x2 + x – 1) (x – 2) + 3
= –x3 + x2 – x + 2x2 – 2x + 2 + 3
= –x3 + 3x2 – 3x + 5
= Dividend
In this way, the division algorithm is verified.