Question

Find the remainder when x^4+x^2-2x^2+1 is divided by x-1

Answer 1

GIVEN :-

• Dividend = x⁴ + x² - 2x² + 1
• Divisor = x - 1

TO FIND :-

• The Remainder.

SOLUTION :-

▪︎Dividend = x⁴ + x² - 2x² + 1 = x⁴ - x² + 1

POLYNOMIAL DIVISION :-

$\boxed{\begin{array}{l | n | r}\bf \: x-1&\sf x^4-x^2+1&\sf x^3+x^2\\ &\sf x^4-x^3\\ & ( - )\:( + )\\&\rule{55}{0.8}\\&\sf\qquad x^3-x^2\\ &\sf\qquad x^3-x^2\\ &\qquad( - )\:\:( + )\\&\quad\rule{60}{0.8}\\&\qquad\qquad\sf1\end{array}}</p><p>$

Hence we found Reamainder = 1

VERIFICATION :-

▪︎Dividend =[ Divisor × Quotient] + Remainder

= [(x - 1)(x³ + x²)] + 1

= [x (x³ + x²) -1 (x³ + x²)] + 1

= x⁴ + x³ - x³ - x² + 1

▪︎x⁴ - x² + 1 = x⁴ - x² + 1

L.H.S = R.H.S

HENCE VERIFIED