Question

Divide 3x^2-x^3-3x+5 by x-1-x^2, and verify the division algorithm.

Answer 1

Question :–

Divide ${3x}^{2} - {x}^{3} - 3x+5$ by $x - 1 - {x}^{2},$ and verify the division algorithm.

Solution :–

$\tt p(x) = {3x}^{2} - {x}^{3} - 3x + 5 = {-x}^{3} + {3x}^{2} - 3x + 5$

$\tt g(x) = x - 1 - {x}^{2} = {-x}^{2} + x + 1$

• Division is refer to the attachment.

By division Algorithm,

$\sf Dividend = Divisor \times Quotient + Remainder$

$\tt p(x) = g(x) \times q(x) + r(x)$

$\implies \tt ({x}^{2} + x - 1) (x-2) + 3$

$\implies \tt {-x}^{3} + {x}^{2} - x + {2x}^{2} - 2x + 2 + 3$

$\implies \tt {-x}^{3} + {3x}^{2} - 3x + 5$

Answer 2

Solution :

$\boxed{\begin{array}{l | n | r}\sf -x^2+x-1&\sf -x^3+3x^2-3x+5&\sf x-2\\ &\sf -x^3+x^2-x\\ & ( + )\:( - )\:( +)\\&\rule{90}{0.8}\\&\sf\qquad 2x^2-2x+5\\ &\sf\qquad 2x^2-2x+2\\ &\qquad( - )\:\:( + )\:\:( - )\\&\quad\rule{90}{0.8}\\&\qquad\qquad\sf 3\end{array}}$

Here,

• Dividend = -x³ + 3x² - 3x + 5

• Divisor = -x² + x - 1

• Quotient = x - 2

• Remainder = 3

Verification :

Division algorithm :-

Dividend = Divisor × Quotient + Remainder

$\bullet\:$-x³ + 3x² - 3x + 5 = (-x² + x - 1) × (x - 2) + 3

$\bullet\:$-x³ + 3x² - 3x + 5 = -x²(x - 2) + x(x - 2) - 1(x - 2) + 3

$\bullet\:$-x³ + 3x² - 3x + 5 = -x³ + 2x² + x² - 2x - x + 2 + 3

$\bullet\:$-x³ + 3x² - 3x + 5 = -x³ + 3x² - 3x + 5

Hence Verified .