Plz cud sombdy show me long division method of x^4/(x-1)(x^2+1)???? Plz note (x-1)(x^2+1) is in the denominator.
Of course! I can help you understand the long division method for dividing a polynomial by another polynomial.
To perform long division for x^4 divided by (x-1)(x^2+1), follow these steps:
Step 1: Write the dividend (x^4) inside the long division symbol (÷).
___________
(x-1)(x^2+1) | x^4
Step 2: Divide the first term of the dividend (x^4) by the first term of the divisor (x).
x^4 ÷ x = x^3
Write this result above the long division symbol.
___________
(x-1)(x^2+1) | x^4
x^3
Step 3: Multiply the divisor (x-1)(x^2+1) by the quotient term x^3.
x^3 * (x-1)(x^2+1) = x^4-x^3+x^2-x^3
Write this product below the dividend.
x^4-x^3
___________
(x-1)(x^2+1) | x^4
x^3
-x^3+x^2
Multiply the divisor (x-1)(x^2+1) by x^3 to get -x^3+x^2.
___________
(x-1)(x^2+1) | x^4
x^3
-x^3+x^2
Step 4: Subtract the product obtained in the previous step from the dividend.
(x^4 - x^3 + x^2) - (-x^3 + x^2) = x^4 + 2x^2
Write the result below the product.
x^4 + 2x^2
_______________
(x-1)(x^2+1) | x^4
x^3
-x^3+x^2
x^4 + 2x^2
Step 5: Repeat the process with the new result (x^4 + 2x^2).
Bring down the next term, which is 0.
x^4 + 2x^2
_______________
(x-1)(x^2+1) | x^4
x^3
-x^3+x^2
x^4 + 2x^2
0
Step 6: Divide the new term (0) by the first term of the divisor (x).
0 ÷ x = 0
Write this result above the line.
x^4 + 2x^2
_______________
(x-1)(x^2+1) | x^4
x^3
-x^3+x^2
x^4 + 2x^2
0
0
Step 7: Multiply the divisor (x-1)(x^2+1) by the quotient term 0.
0 * (x-1)(x^2+1) = 0
Write this product below the line.
x^4 + 2x^2
_______________
(x-1)(x^2+1) | x^4
x^3
-x^3+x^2
x^4 + 2x^2
0
0
Step 8: Subtract the product obtained in the previous step from the previous result (0).
(0 - 0) = 0
Write the result below the product.
x^4 + 2x^2
_______________
(x-1)(x^2+1) | x^4
x^3
-x^3+x^2
x^4 + 2x^2
0
0
0
Step 9: Since there are no more terms to bring down, the division process is complete.
The final result is x^3 - x + 2 with a remainder of 0.
Therefore, the long division of x^4 divided by (x-1)(x^2+1) is x^3 - x + 2.
I hope this helps! Let me know if you have any further questions.