Eight Standard >> Division of a polynomial by another polynomial | Part - 1

Division of a polynomial by another polynomial 

How to divide monomial by a monomial?

To divide a monomial by another monomial, you can use the concept of division in algebra. The general rule for dividing monomials is to divide their numerical coefficients and then divide their variables' exponents.

Here's the step-by-step process to divide one monomial by another:

Step 1: Identify the numerical coefficients of both monomials.
A monomial is an expression of the form: \(c * x^n\), where 'c' is the numerical coefficient and 'x' is the variable raised to the exponent 'n'. For example, in the monomial \(3x^2\), the numerical coefficient is 3.

Step 2: Divide the numerical coefficients.
Take the numerical coefficient of the dividend (the monomial being divided) and divide it by the numerical coefficient of the divisor (the monomial you are dividing by). For example, if you have \(\frac{6x^3}{2x^2}\), divide 6 by 2 to get 3.

Step 3: Divide the variables' exponents.
Divide the exponent of the variable in the dividend by the exponent of the variable in the divisor. For example, if you have \(\frac{6x^3}{2x^2}\), divide 3 by 2 to get \(x^(3-2)\) = \(x^1\) = x.

Step 4: Write the simplified result.
Combine the results from Step 2 and Step 3. In our example, you would get 3x as the simplified result.

So, \(\frac{6x^3}{2x^2}\) simplifies to 3x. That's how you divide one monomial by another!

How to divide binomial or trinomial by a monomial?

To divide a binomial or trinomial by a monomial, you can use the concept of long division in algebra. The process is similar to dividing numbers using long division. Here's how you can do it step-by-step:

Let's start with an example of dividing a binomial by a monomial:

Example: \({(4x^2 + 6x)}{2x}\)

Step 1: Write the division problem in the long division format.
__________
2x | \(4x^2 + 6x\)

Step 2: Divide the first term of the dividend by the divisor.
\(\frac{4x^2}{2x}\) = 2x

Step 3: Write the result from Step 2 above the division symbol.
__________
2x | \(4x^2 + 6x\)
2x

Step 4: Multiply the divisor (2x) by the result (2x) and write the product below the first term of the dividend.
__________
2x | \(4x^2 + 6x\)
2x

Step 5: Subtract the result from Step 4 from the first term of the dividend.
__________
2x | \(4x^2 + 6x\)
2x

Step 6: Bring down the next term from the dividend (6x) and write it next to the result obtained in Step 2.
__________
2x | \(4x^2 + 6x\)
2x + 0

Step 7: Divide the new expression (6x) by the divisor (2x).
6x / 2x\) = 3

Step 8: Write the result from Step 7 above the division symbol.
__________
2x | \(4x^2\) + 6x
2x + 3

Step 9: Check if there are any terms left to bring down. Since there are none, the division is complete.

The final result is: \(\frac{(4x^2 + 6x)}{2x}\) = 2x + 3.

Division of a polynomial by another polynomial in the form of P(x)/Q(x), Provided degree of P(x) > degree of Q(x) using long devision method

To divide a polynomial P(x) by another polynomial Q(x) using long division, follow these steps:

Step 1: Arrange the polynomials properly.
Make sure that both P(x) and Q(x) are written in descending order of their degrees (highest degree to lowest degree). If there are any missing terms, include them with coefficients of zero.

Step 2: Identify the leading terms.
The leading term of P(x) is the term with the highest degree, and the leading term of Q(x) is the term with the highest degree in the divisor.

Step 3: Divide the leading term of P(x) by the leading term of Q(x).
This division will give you the first term of the quotient.

Step 4: Write the first term of the quotient above the long division symbol.
This is the term obtained in Step 3.

Step 5: Multiply the entire divisor Q(x) by the first term of the quotient.
This multiplication will give you a new polynomial.

Step 6: Subtract the polynomial obtained in Step 5 from the original polynomial P(x).
This will give you a new polynomial.

Step 7: Repeat the process.
Now, you have a new polynomial obtained in Step 6. Repeat the process from Step 2 using this new polynomial as the dividend, and continue the process until the degree of the new polynomial is less than the degree of Q(x).

Step 8: Write the final result.
Once you can no longer divide, the remaining polynomial is the remainder, and the terms above the long division symbol form the quotient.

Example:
Let's say we want to divide P(x) = \(4x^3\) - \(3x^2\) + \(2x\) - 6 by Q(x) = 2x - 1.

Step 1: Arrange the polynomials:
________________
2x - 1 | \(4x^3\) - \(3x^2\) + 2x - 6

Step 2: Identify the leading terms:
Leading term of P(x): \(4x^3\)
Leading term of Q(x): \(2x\)

Step 3: Divide the leading terms:
\(\frac{4x^3}{2x}\) = \(2x^2\)

Step 4: Write the first term of the quotient:
________________
2x - 1 | \(4x^3\) - \(3x^2\) + 2x - 6
\(2x^2\)

Step 5: Multiply the entire divisor by the first term of the quotient:
\(2x^2 * (2x - 1)\) = \(4x^3\) - \(2x^2\)

Step 6: Subtract the result from Step 5 from the original polynomial:
\(4x^3\) - \(3x^2\) + 2x - 6 - \((4x^3 - 2x^2)\) = -\(x^2\) + 2x - 6

Step 7: Repeat the process:
Now we have a new polynomial -\(x^2\) + 2x - 6. The degree of this polynomial (2) is less than the degree of Q(x) (1), so we stop here.

Step 8: Write the final result:
The quotient is \(2x^2\), and the remainder is -\(x^2\) + 2x - 6.

The final result of dividing P(x) by Q(x) is: P(x)/Q(x) = \(2x^2\) - \(\frac{(x^2 - 2x + 6)}{(2x - 1)}.