on dividing x3 - 3x2 +x + 2 by a polynomial g(x) the quotient and reminder were x - 2 and -2 + 4 respectively. find g(x)
Answers
Given :- On dividing x³ - 3x² + x + 2 by a polynomial g(x) the quotient and reminder were x - 2 and -2x + 4 respectively. find g(x) = ?
Solution :-
we have,
→ Dividend = x³- 3x² + x + 2
→ Divisor = g(x)
→ Quotient = (x - 2)
→ Remainder = (-2x + 4)
we know that,
- Dividend = Divisor * Quotient + Remainder
putting values we get,
→ x³- 3x² + x + 2 = g(x) * (x - 2) + (-2x + 4)
→ x³- 3x²+ x + 2 - 4 + 2x = g(x) * (x - 2)
→ x³ - 3x² + 3x - 2 = g(x) * (x - 2)
→ x³ - 3x² + 2x + x - 2 = g(x) * (x - 2)
→ x(x² - 3x + 2) + x - 2 = g(x) * (x - 2)
→ x(x² - 2x - x + 2) + x - 2 = g(x) * (x - 2)
→ x[x(x - 2) - 1(x - 2)] + (x - 2) = g(x) * (x - 2)
→ x[(x - 2)(x - 1)] + (x - 2) = g(x) * (x - 2)
→ (x - 2)[x(x - 1) + 1] = g(x) * (x - 2)
→ x(x - 1) + 1 = g(x)
→ (x² - x + 1) = g(x)
therefore,
→ g(x) = (x² - x + 1) (Ans.)
Learn more :-
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