Factorization

Factorizing linear expressions involves finding common factors in each term and factoring them out.

Example 1: Factorize the expression $5a + 5b$

Solution: In the expression $5a + 5b$, we can observe that both terms have a common factor of 5. So, we factor out 5 from both terms: $5a + 5b = 5(a + b)$

Example 2: Factorize $18xyz - 24xwz$

Solution: Here, we need to find the highest common factor (HCF) of the two terms. The common factors are 6, $x$, and $z$. So, factoring out $6xz$ gives: $18xyz - 24xwz = 6xz(3y - 4w)$

When a quadratic expression is written as the product of two factors, we say that the expression is factorized. There are different methods to factorize quadratic expressions:

• Factorization by splitting the middle term
• Factorization by inspection

This method involves splitting the middle term (the coefficient of $x$ in $ax^2 + bx + c$) into two terms that can be factored.

Example 1: Factorize $3x^2 - 2x - 8$ by splitting the middle term

Solution: We start with the quadratic expression $3x^2 - 2x - 8$. We need to find two numbers that multiply to $3 \times (-8) = -24$ and add up to -2. The numbers are 4 and -6, since $4 \times -6 = -24$ and $4 + (-6) = -2$. $3x^2 - 2x - 8 = 3x^2 + 4x - 6x - 8$ Now, factor by grouping: $= x(3x + 4) - 2(3x + 4)$ Factor out the common binomial factor: $= (3x + 4)(x - 2)$

Example 2: Factorize $x^2 + 10x + 25$ by splitting the middle term

Solution: The product of $1 \times 25 = 25$, and the two numbers that multiply to 25 and add up to 10 are 5 and 5. So, we split the middle term as follows: $x^2 + 10x + 25 = x^2 + 5x + 5x + 25$ Now, factor by grouping: $= x(x + 5) + 5(x + 5)$ Factor out the common binomial factor: $= (x + 5)(x + 5) = (x + 5)^2$

In this method, we look at the quadratic expression and directly identify the factors based on the coefficients.

Example 1: Factorize $x^2 + 3x + 2$ by inspection

Solution: We need two numbers that multiply to $1 \times 2 = 2$ and add up to 3. The numbers are 1 and 2, so we can directly write the factors as: $x^2 + 3x + 2 = (x + 1)(x + 2)$

Example 2: Factorize $4x^2 + 5x - 6$ by inspection

Solution: We need two numbers that multiply to $4 \times (-6) = -24$ and add up to 5. The numbers are 8 and -3. We split the middle term: $4x^2 + 5x - 6 = 4x^2 + 8x - 3x - 6$ Now, factor by grouping: $= 4x(x + 2) - 3(x + 2)$ Factor out the common binomial factor: $= (x + 2)(4x - 3)$