Please use systemic division. (x^4-6x^2+3x-1) / (x-2) please help thank you
To perform long division using the systemic division method, you can follow these steps:
Step 1: Write the dividend and divisor in descending exponent order. In this case, the dividend is x^4 - 6x^2 + 3x - 1, and the divisor is x - 2. Rewrite the terms in descending exponent order (if needed) as: x^4 + 0x^3 - 6x^2 + 3x - 1 divided by x - 2.
Step 2: Divide the leading term of the dividend by the leading term of the divisor. In this case, x^4 / x = x^3. Write this as the first term of the quotient above the division bar.
Step 3: Multiply the divisor by the first term of the quotient and write the result below the dividend, aligning like terms.
x - 2
_______________
x^4 + 0x^3 - 6x^2 + 3x - 1
x^4 - 2x^3
Step 4: Subtract the result obtained in Step 3 from the dividend.
x^4 + 0x^3 - 6x^2 + 3x - 1
- (x^4 - 2x^3)
__________________________
2x^3 - 6x^2 + 3x - 1
Step 5: Bring down the next term of the dividend (3x) and repeat Steps 2 to 4.
Step 6: Divide the leading term of the new dividend (2x^3) by the leading term of the divisor (x). In this case, 2x^3 / x = 2x^2. Write this as the next term of the quotient above the division bar.
Step 7: Multiply the divisor by the new term of the quotient (2x^2) and write the result below the current dividend, aligning like terms.
x - 2
_______________
x^4 + 0x^3 - 6x^2 + 3x - 1
- (x^4 - 2x^3)
__________________________
2x^3 - 6x^2 + 3x - 1
- (2x^3 - 4x^2)
________________________
-2x^2 + 3x - 1
Step 8: Repeat Steps 5 to 7 until you have gone through all the terms of the dividend.
Step 9: In the end, you will have a remainder or a quotient equal to zero. The quotient is obtained by combining all the terms above the division bar.
In this case, the final quotient is x^3 + 2x^2 - 2, and the remainder is -2x^2 + 3x - 1.