Factorization & Simplification

Simplification in Sum and Difference Expressions

In sum and difference expressions, simplification can be achieved by combining like terms. By merging identical terms, calculations become more straightforward.

Simplification of Fractions

Simplifying fractions involves dividing both the numerator and the denominator by their greatest common divisor. Let’s break it down step by step:

Example :

Simplify the fraction:

    \[\frac{18}{24}\]

First, find the GCD of the numerator and denominator, which is 6:

    \[18 \div 6 = 3 \quad \text{and} \quad 24 \div 6 = 4\]

Thus, the simplified fraction is:

    \[\frac{18}{24} = \frac{3}{4}\]

Here, both the numerator and denominator were divided by 6, resulting in a simpler fraction.

Simplification in Equations

Simplification in equations involves dividing both sides of the equation by the same number. It is important to apply the same operation to both sides to maintain the equality.

Example :

Simplify the equation:

    \[ \frac{10}{15} = \frac{4}{6}\]

We simplify each fraction individually.

First fraction:

    \[\frac{10}{15} = \frac{10 \div 5}{15 \div 5} = \frac{2}{3}\]

Second fraction:

    \[\frac{4}{6} = \frac{4 \div 2}{6 \div 2} = \frac{2}{3}\]

Thus, the equality is verified:

    \[\frac{10}{15} = \frac{4}{6} = \frac{2}{3}\]

Simplification Between Denominators in Proportions

In proportional expressions, simplification can also be performed between denominators. This makes it easier to compare the ratios. When simplifying denominators, the relationships between the numerator and denominator must be considered.

Example :

Simplifying a Ratio:

    \[\frac{16}{24} : \frac{8}{12}\]


First, simplify the numerator and denominator of the first ratio:

    \[\frac{16}{24} = \frac{16 \div 8}{24 \div 8} = \frac{2}{3}\]

Next, simplify the numerator and denominator of the second ratio:

    \[\frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3}\]

Finally, the simplified ratios become:

    \[\frac{2}{3} : \frac{2}{3}\]

Simplification of a More Complex Expression

Example :

Simplify the expression given:

    \[\frac{x^2 + 5x + 6}{x + 2}\]

Step 1: Factorize the numerator:

    \[x^2 + 5x + 6 = (x + 2)(x + 3)\]

Step 2: Simplify by canceling the common factor:

    \[\frac{(x + 2)(x + 3)}{x + 2} = x + 3\]

Simplification of a More Challenging Expression

Example :

Simplify the fraction given:

    \[\frac{3x^2 - 12x}{6x^2 - 24x}\]

Step 1: Factorize both the numerator and the denominator:

    \[3x^2 - 12x = 3x(x - 4)\]


    \[6x^2 - 24x = 6x(x - 4)\]

Step 2: Simplify by canceling the common factors:

    \[\frac{3x(x - 4)}{6x(x - 4)} = \frac{3}{6} = \frac{1}{2}\]

Result:

    \[\frac{1}{2}\]

Expansion

Expansion is the process of multiplying both the numerator and denominator of a fraction by the same positive or negative number. This operation does not change the value of the fraction; only the terms in the numerator and denominator are increased. In other words, expanding a fraction does not alter the number it represents but changes its appearance.

Expansion is used to express fractions in different forms. For example, a fraction can be represented with larger or smaller terms, but the ratio between the numerator and denominator remains unchanged. This is particularly useful when combining fractions with common denominators or when comparing fractions.

Example of Expansion:

Consider the fraction 2/3 If we expand this fraction by multiplying it by 2:

(1)   \begin{equation*}\frac{3}{2} \times \frac{2}{2} = \frac{6}{4}\end{equation*}

Here, both the numerator and the denominator have been multiplied by the same number (2), and the value of the fraction has not changed

Factoring Identities

Methods of Factoring

Common Factor Method:

This method is used when there is a common factor in every term. The common factor is taken out of the parentheses to simplify or factor the expression.

Example:

In the expression 3x + 6, the common factor in both terms is 3. Therefore, the expression can be written as:

3x + 6 = 3(x + 2)

The Common Factor Method is a fundamental technique used especially for simplifying or factoring expressions.

Grouping Method:

This method is used when there is no common factor in each term. Instead, the terms are grouped into pairs (or sometimes threes), and the common factor within each group is factored out. After grouping, common terms are factored out to complete the process.

Example:

In the expression y2 + 7y + 3y + 12, we group the terms as:

(y2 + 7y) + (3y + 12)

Now, factor out the common factors from each group:

y(y + 7) + 3(y + 4)

Finally, factor out the common term (y + 4):

(y + 4)(y + 3)

The Grouping Method is an effective technique for factoring expressions that do not have a simple common factor across all terms.

Using Identities in Factoring:

Identities are expressions that follow a specific pattern or structure that can be used to simplify the factoring process. By recognizing these identities, factoring becomes quicker and easier.

Example:

The expression z2 – 16 follows the difference of squares identity, which states:

a2 – b2 = (a + b)(a – b)

Therefore, we can factor z2 – 16 as:

z2 – 16 = (z + 4)(z – 4)

Using these types of identities, more complex expressions can be quickly factored.

Rational Expressions:

Rational expressions are expressions where the numerator and the denominator are polynomials. These expressions can be simplified and manipulated similarly to rational numbers. A general form of a rational expression is:

    \[\frac{P(x)}{Q(x)}\]

Simplifying Rational Expressions:

To simplify a rational expression, both the numerator and the denominator are factored, and any common factors are cancelled out.

Example:

For the expression 8y / 12, we simplify it as follows:

    \[\frac{8y}{12} = \frac{4 \cdot 2y}{4 \cdot 3} = \frac{2y}{3}\]

Here, the number 4 is the common factor of both the numerator and denominator, so it is cancelled out to simplify the expression.

17 Jan, 2025