How to Solve Algebraic Long Division Method?

Such algebraic expressions which are made up of coefficients and variables are called polynomials. The general form of a polynomial is given as:

a_nxⁿ + a_n−1xⁿ⁻¹ + … + a₂x² + a₁x + a₀

Polynomials can be classified as monomials, binomials and trinomials based upon the number of terms present. For example, the terms like x, 13y, 39, etc. are all monomials while terms like x² + x, x¹⁰ – x⁴, etc. are termed as binomials because they consist of two terms. Similarly, such polynomials as having only three terms are termed as trinomials.

Long Division of Polynomials

The long division method is the most frequent and general method for dividing polynomials by binomials or any other form of polynomials. In case the given numerator and denominator do not have any common factors, you can simplify the expression by using the long division method. An example of the long division method is shown as follows, followed by the steps:

Suppose we are asked to divide the polynomial x² + 2x + 3 by x – 2 by long division. It is performed below:

The quotient and remainder are x + 4 and 11 respectively.

Steps to Solve

Step 1: Arrange the terms of the given polynomial in the decreasing order of their powers. In the given question, the polynomial need not be arranged. Use zero as the coefficient of the missing terms.

Step 2: Divide the dividend’s first term (x²) by the divisor’s first term, and use it as the quotient’s first term.

Step 3: Multiply the divisor (x – 2) in this case by the answer (x), then place the product beneath the dividend(x² + 2x + 3).

Step 4: Subtract the product from the dividend and write the answer beneath by drawing a line. 

Step 5: Use the new polynomial obtained after subtraction to repeat the operation, i.e., divide that polynomial by the divisor and write it as the quotient’s second term.

Keep repeating the above steps until no further division can take place. Note down the quotient and remainder.

Sample Problems

Question 1. Solve  using long division.

Solution:

Dividend = 6x⁴ + 0x³ + 9x² + 3x + 6

Divisor = x² – 2

Using long division method, we have:

Thus, the quotient and remainder are 6x² + 3 and 3 respectively.

Question 2. Is 6x³ + 12x² + 2x + 25 completely divisible by x² + 4x + 3?

Solution:

Dividend = 6x³ + 12x² + 2x + 25

Divisor = x² + 4x + 3

Using long division method, we have:

Since the remainder is non- zero, the divisor (x² + 4x + 3) is not a factor of the dividend (6x³ + 12x² + 2x + 25).

Question 3. Is 6x² – 4x – 24 completely divisible by x – 3?

Solution:

Dividend = 6x² – 4x – 24

Divisor = x – 3

Using long division method, we have:

Since the remainder is non- zero, the dividend is not completely divisible by the divisor.

Question 4. Find the quotient and remainder of  x³ – x² + x – 1 and 2x + 1 using long division method.

Solution:

Dividend = x³ – x² + x – 1

Divisor = 2x + 1

Using long division method, we have:

Thus, the quotient and remainder are and respectively.

Question 5. Solve (18x⁴+9x³+3x²)÷(3x²+1) using long division.

Solution:

Dividend = 18x⁴ + 9x³ + 3x² + 0x + 0

Divisor = 3x² + 1

Using long division method, we have:

Thus, the quotient and remainder are 6x² + 3x – 1 and -3x + 1 respectively.

Question 6. Solve  using long division.

Solution:

Dividend = 4x³ + 5x² + 5x + 8

Divisor = 4x + 1

Using long division method, we have:

Thus, the quotient and remainder are x² + x + 1 and 7 respectively.