Division of Polynomials or algebraic expressions | Basic of Algebra | MathOguide

Division of Polynomials or algebraic expressions | Basic of Algebra

Steps for Long Division of Polynomials:

Let's take a general polynomial division example:

Divide by .

Step-by-Step Solution:

Step 1: Set up the division.

Dividend:

Divisor:


Start with the division symbol:

\frac{3x^3 + 5x^2 - 2x + 7}{x + 2}

Step 2: Divide the first term of the dividend by the first term of the divisor.

Divide by , which gives .

Now, write as the first term of the quotient.

Step 3: Multiply the divisor by the first term of the quotient.

Multiply by :

3x^2 \cdot (x + 2) = 3x^3 + 6x^2

Step 4: Subtract this result from the dividend.

Now subtract from :

(3x^3 + 5x^2 - 2x + 7) - (3x^3 + 6x^2) = -x^2 - 2x + 7

Step 5: Repeat the process with the new polynomial.

Now, divide the first term of the new polynomial by , which gives .

Write as the next term in the quotient.

Step 6: Multiply the divisor by the new term of the quotient.

Multiply by :

-x \cdot (x + 2) = -x^2 - 2x

Step 7: Subtract this result from the current polynomial.

Subtract from :

(-x^2 - 2x + 7) - (-x^2 - 2x) = 7

Step 8: Repeat the process one last time.

Now divide the constant term by , but since is a constant and cannot be divided by , it becomes the remainder.

Thus, the quotient is and the remainder is .

Final Answer:

\frac{3x^3 + 5x^2 - 2x + 7}{x + 2} = 3x^2 - x + \frac{7}{x + 2}


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