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#### Exercise 12.3 class 8 solution-Division of algebric expression

Division of algebraic expressions is a fundamental operation in algebra, and it involves dividing one algebraic expression by another. To divide algebraic expressions, you can follow these steps:

**Factor if possible:** Before you begin division, it’s often helpful to factor both the numerator and denominator to simplify the expressions. This step is not always necessary, but it can make the division process easier.

**Invert and Multiply:** To divide one algebraic expression by another,you can multiply the dividend (the expression you’re dividing into) by the reciprocal of the divisor. The reciprocal of an expression is obtained by flipping the numerator and the denominator.

For example, if you have the expression A divided by B, you can rewrite it as:

A / B = A * (1/B)

Where 1/B is the reciprocal of B.

**Multiply:** Now, simply multiply the dividend by the reciprocal of the divisor, just like you would in regular multiplication. You can also apply the distributive property if there are multiple terms involved.

Example: Suppose you want to divide (2x^2 + 3x) by (x + 1). First, find the reciprocal of (x + 1), which is 1/(x + 1). Then, multiply the dividend by the reciprocal:

(2x^2 + 3x) / (x + 1) = (2x^2 + 3x) * (1/(x + 1))

You can distribute the reciprocal to both terms in the dividend:

= (2x^2 * 1/(x + 1)) + (3x * 1/(x + 1))

**Simplify:** If possible, simplify the resulting expression by canceling common factors or simplifying further.

Example: Continuing from the previous example:

= (2x^2 * 1/(x + 1)) + (3x * 1/(x + 1)) = (2x^2 / (x + 1)) + (3x / (x + 1))

- You could factor x out of each term in the denominators:= (2x(x) / (x + 1)) + (3x / (x + 1))Now you have a common denominator, and you can add the two fractions:= (2x^2 + 3x) / (x + 1)

So, the result of dividing (2x^2 + 3x) by (x + 1) is (2x^2 + 3x) / (x + 1). This is the simplified expression after division.

Dividing a monomial by another monomial is a straight forward process in algebra. Monomial are expressions that consist of a single term, such as “3x,” “2y^2,” or “4.”

Here’s how you can divide one monomial by another:

1.**Write the expression:** Start with the expression you want to divide, which is often written in the form of (a monomial) / (another monomial).

2.**Apply the division rule:** To divide one monomial by another, you apply the rule for dividing like bases. If both monomial have the same base (variable), you can simply subtract the exponents.

For example, if you have “2x^3” divided by “x,” you can subtract the exponents:

3 – 1 = 2.

Example: Divide “6x^4” by “2x^2.” Both monomial have the same base “x,” so you subtract the exponents:

(6x^4) / (2x^2) = (6/2) * (x^4/x^2) = 3x^(4-2) = 3x^2

3.**Simplify:** In most cases, you should simplify the result. In the example above, “3x^2” is already in its simplest form. If there are common factors between the coefficients or if you can simplify further, do so.

Example: Divide “12x^5” by “4x^3.”

Start by simplifying the coefficients and then subtract the exponents:(12x^5) / (4x^3) = (12/4) * (x^5/x^3) = 3x^(5-3) = 3x^2

In this case, the coefficient 12 and 4 have a common factor of 4, so you can simplify it further:(12x^5) / (4x^3) = (3x^2)

So, when you divide a monomial by another monomial, you simplify the coefficients and apply the rule for dividing like bases by subtracting the exponents. This results in a simplified monomial

Example 1.divide 12x^5 by -4x^3.

#### Exercise 12.3 class 8 solution-Dividing a polynomial by monomial

Dividing a polynomial by a monomial is also a common operation in algebra. To do this, you apply the distributive property to divide each term in the polynomial by the monomial. Here are the steps to divide a polynomial by a monomial:

**Write the expression:**Start with the expression you want to divide, which is usually in the form of (a polynomial) / (a monomial).**Apply the distributive property:**Divide each term in the polynomial by the monomial. This means you’ll divide each term’s coefficient by the monomial’s coefficient (if applicable) and subtract the exponents of the variables (if the base variable matches the monomial.- Example:
- Divide the polynomial “6x^3 – 12x^2 + 18x” by the monomial “3x.” You’ll divide each term in the polynomial by “3x”:
- (6x^3 – 12x^2 + 18x) / (3x)
- for the first term, “6x^3,” you divide the coefficient 6 by 3 and subtract the exponents: (6x^3) / (3x) = (6/3) * (x^3/x) = 2x^(3-1) = 2x^2
- For the second term, “-12x^2,” divide the coefficient -12 by 3, and subtract the exponents: (-12x^2) / (3x) = (-12/3) * (x^2/x) = -4x^(2-1) = -4x
- For the third term, “18x,” divide the coefficient 18 by 3, and subtract the exponents: (18x) / (3x) = (18/3) * (x/x) = 6x^(1-1) = 6

**Combine the results:**After dividing each term, you can combine the results into a simplified expression.So, in this example, the result of dividing “6x^3 – 12x^2 + 18x” by “3x” is:(6x^3 – 12x^2 + 18x) / (3x) = 2x^2 – 4x + 6- The simplified expression “2x^2 – 4x + 6” is the result of dividing the polynomial by the monomial “3x.”

**Example : Divide 6x^4+24x^3-5x^2 by 3x^2 **