MULTIPLICATION & DIVISION OF ALGEBRAIC EXPRESSION OR POLYNOMIALS

MULTIPLICATION & DIVISION OF ALGEBRAIC EXPRESSION OR POLYNOMIALS
division of an algebraic expression
What is Division of Algebraic Expressions ? 
    In the division of an algebraic expression, we cancel the common terms. which is similar to the division of the numbers. Division of algebraic expressions involves the following steps.
    Step 1: Directly take out common terms of factories the given expressions to check for the common terms.
    Step 2: Cancel the common term.
Note: Here, the common terms correspond to either of the following : constants, variables, terms or just coefficients.
Division of algebraic expressions
    \(\left( {{a^2} + a} \right) \div a\) = a (a + 1) / a (take out common terms)
               = a + 1 (cancel out common terms)
    Thus, \(\left( {{a^2} + a} \right) \div a = a + 1\)
There are different types of division of algebraic expressions.
1. Division of monomial by a monomial
2. Division of polynomial by a monomial
3. Division of polynomial by a Polynomial (Binomial)
4. Division of polynomial by a Polynomial (Trinomial)
In any case, we first take out common terms from the given polynomials and then eliminate that common term/terms. Let us discuss them case by case.
1) Division of Monomial by a Monomial
    A monomial is a type of expression that has  only one term. The correct method to perform the division of monomial by another monomial is given below:
Example : \(\frac{{ - 8{a^3}bc}}{{2ab}} = \frac{{ - 8 \times a \times a \times a \times b \times c}}{{2 \times a \times b}} = - 4{a^2}c\)
2) Division of Polynomial by a Monomial
    Dividing a polynomial by a monomial involves dividing each term of the polynomial by the monomial. Here are the steps:
Step 1: Write the polynomial and the monomial.
    Let's say you have the polynomial P(x) and the monomial M(x):
    \(P(x) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + ...... + {a_1}x + {a_0}\)
    \(M(x) = b{x^k}\)
Step 2: Identify each term in the polynomial.
    In the polynomial \(P(x) = {a_n}{x^n}\) is the term with the highest power of x , and is the constant term
Step 3: Divide each term of the polynomial by the monomial.
    Divide each term of P(x) by M(x)
    \(P(x) = \frac{{{a_n}{x^n}}}{{b{x^k}}} + \frac{{{a_{n - 1}}{x^{n - 1}}}}{{b{x^k}}} + ...... + \frac{{{a_1}x}}{{b{x^k}}} + \frac{{{a_0}}}{{b{x^k}}}\)
Step 4: Simplify each term.
    Reduce each fraction if possible, simplifying the expression:
    \({a_n}{x^{n - k}} + \frac{{{a_{n - 1}}}}{b}{x^{(n - 1) - k}} + ...... + \frac{{{a_1}}}{b}{x^{1 - k}} + \frac{{{a_0}}}{b}{x^{ - k}}\)
Step 5: Write the final answer.
    Combine like terms and rewrite the expression:
    \(Q(x) = {a_n}{x^{n - k}} + \frac{{{a_{n - 1}}}}{b}{x^{(n - 1) - k}} + ...... + \frac{{{a_1}}}{b}{x^{1 - k}} + \frac{{{a_0}}}{b}{x^{ - k}}\)
    This is the result of dividing the polynomial P(x) by a monomial M(x). . The resulting quotient is 
A polynomial contains a few types of expressions, some of which are binomial, trinomial.    
    Now, let’s perform the dividing polynomials by monomials.
\(\left( {4{y^3} + 5{y^2} + 6y} \right) \div 2y\)=\(\frac{{2y\left( {2{y^2} + \frac{5}{2}y + 3} \right)}}{{2y}} = 2{y^2} + \frac{5}{2}y + 3\)
3) Dividing Polynomial By a Binomial
    Dividing a polynomial by a binomial involves a series of steps similar to long division. Here's a step-by-step guide:
Step 1: Write the polynomial in standard form. Arrange the terms in descending order of the degree.
Step 2: Write the binomial divisor, placing it on the left side of the division symbol.
Step 3: Divide the first term of the polynomial by the first term of the divisor. Write the result on top, above the division symbol.
Step 4: Multiply the entire divisor by the result obtained in Step 3, and write the result below the polynomial, aligning like terms.
Step 5: Subtract the result obtained in Step 4 from the polynomial.
Step 6: Bring down the next term from the original polynomial.
Step 7: Repeat Steps 3-6 until you have brought down all terms of the polynomial.
Step 8: The quotient obtained above the division symbol is the final quotient, and any remaining terms or constants constitute the remainder.
Here's an example to illustrate the steps:
    Divide 3x³+5x²-2x-4 by x-2.

Therefore, 3x³+5x²-2x-4 divided by x-2 equals 3x² + 11x + 22 with a remainder of 36
Example in another simple way : 
Solution : In numerator 7x is common, take out 7x
\(\frac{{7{x^2} + 14x}}{{\left( {x + 2} \right)}} = \frac{{7x\left( {x + 2} \right)}}{{\left( {x + 2} \right)}} = 7x\)
4) Dividing Polynomial By a Trinomial
Dividing a polynomial by a trinomial involves a similar process to polynomial division but with an additional degree of complexity. The steps are akin to long division, and here's how you can divide a polynomial by a trinomial:
Step 1: Arrange the polynomials
    Write both the dividend (numerator) and the divisor (trinomial denominator) in long division form. Make sure the terms are arranged in descending order of degrees.
Step 2: Divide the leading term
    Divide the leading term of the dividend by the leading term of the divisor. Write the result above the division bar.
Step 3: Multiply the divisor by the result
    Multiply the entire trinomial divisor by the result obtained in Step 2. Write the result below the dividend, aligning like terms.
Step 4: Subtract
    Subtract the result obtained in Step 3 from the dividend.
Step 5: Bring down the next term
    Bring down the next term from the dividend.
Step 6: Repeat
    Repeat steps 2 to 5 until you have brought down all the terms of the dividend.
Step 7: Write the final answer
The quotient is the result obtained from the division, and any remainder (if it exists) is written as a fraction with the divisor as the denominator.
Here's a simplified example
Divide 3x³+4x²-5x+1 by x²-2x+1

So, 3x³+4x²-5x+1 divided by x²-2x+1 is 3x + 6 with a remainder of 12x-9
Remember to verify your answer by multiplying the quotient with the divisor and adding the remainder to see if you get the original dividend.