Division of Polynomials: 6x^4+5x^3+3x-5 divided by 3x^2-2x
In this article, we will discuss the division of polynomials, specifically the problem of dividing 6x^4+5x^3+3x-5 by 3x^2-2x.
The Division Algorithm
The division algorithm is a process of dividing a polynomial by another polynomial. It is similar to the long division algorithm used in arithmetic, but with polynomials.
Step-by-Step Solution
To divide 6x^4+5x^3+3x-5 by 3x^2-2x, we will follow the division algorithm:
Step 1: Write the dividend and divisor
Dividend: 6x^4+5x^3+3x-5 Divisor: 3x^2-2x
Step 2: Divide the leading term of the dividend by the leading term of the divisor
Leading term of dividend: 6x^4 Leading term of divisor: 3x^2 Quotient: 2x^2
Step 3: Multiply the quotient by the divisor and subtract the product from the dividend
(2x^2)(3x^2-2x) = 6x^4-4x^3 Subtract: 6x^4+5x^3+3x-5 - (6x^4-4x^3) = 9x^3+3x-5
Step 4: Repeat steps 2 and 3 with the new dividend
New dividend: 9x^3+3x-5 Divisor: 3x^2-2x
Step 5: Divide the leading term of the new dividend by the leading term of the divisor
Leading term of new dividend: 9x^3 Leading term of divisor: 3x^2 Quotient: 3x
Step 6: Multiply the quotient by the divisor and subtract the product from the new dividend
(3x)(3x^2-2x) = 9x^3-6x^2 Subtract: 9x^3+3x-5 - (9x^3-6x^2) = 6x^2+3x-5
Step 7: Repeat steps 2 and 3 with the new dividend
New dividend: 6x^2+3x-5 Divisor: 3x^2-2x
Step 8: Divide the leading term of the new dividend by the leading term of the divisor
Leading term of new dividend: 6x^2 Leading term of divisor: 3x^2 Quotient: 2
Step 9: Multiply the quotient by the divisor and subtract the product from the new dividend
(2)(3x^2-2x) = 6x^2-4x Subtract: 6x^2+3x-5 - (6x^2-4x) = 7x-5
Since the new dividend is a linear polynomial, we can stop here.
Final Answer
The final answer is:
Quotient: 2x^2 + 3x + 2 Remainder: 7x-5
Therefore, we have:
6x^4+5x^3+3x-5 = (2x^2 + 3x + 2)(3x^2-2x) + 7x-5
