Factorization by Splitting the Middle Term

Factorization by splitting the middle term is a method used to factorize quadratic expressions of the form:

ax2+bx+cax^2 + bx + cax2+bx+c

where a, b, and c are constants, and b is the middle term coefficient.

This method works by splitting the middle term into two terms whose coefficients add up to b and whose product equals a × c.

Steps for Factorization by Splitting the Middle Term

Step 1: Identify the Coefficients

In the given quadratic expression ax² + bx + c, identify the values of a, b, and c.

Step 2: Find Two Numbers

Find two numbers that:

• Multiply to give the product of a × c
• Add to give the middle term coefficient b

Step 3: Split the Middle Term

Rewrite the middle term bx as the sum of two terms using the numbers found in Step 2.

Step 4: Factor by Grouping

Group the terms in pairs and take the common factors out.

Step 5: Factor Out the Common Binomial

After grouping, take out the common binomial factor to get the final factorized expression.

Examples

Example 1: Factorize x² + 7x + 10

Step 1: Identify the coefficients

• Here, a = 1, b = 7, c = 10
• The product a × c = 1 × 10 = 10

Step 2: Find two numbers that multiply to 10 and add to 7

• The numbers 2 and 5 satisfy this condition because 2 × 5 = 10 and 2 + 5 = 7

Step 3: Split the middle term

x2+2x+5x+10x² + 2x + 5x + 10x2+2x+5x+10

Step 4: Factor by grouping

(x2+2x)+(5x+10)(x² + 2x) + (5x + 10)(x2+2x)+(5x+10)

Taking out common factors:

$$x(x+2)+5(x+2)$$

Step 5: Factor out the common binomial factor

$$(x+2)(x+5)$$

Thus, x² + 7x + 10 = (x + 2)(x + 5)

Example 2: Factorize 2x² + 11x + 15

Step 1: Identify the coefficients

• Here, a = 2, b = 11, c = 15
• The product a × c = 2 × 15 = 30

Step 2: Find two numbers that multiply to 30 and add to 11

• The numbers 5 and 6 satisfy this condition because 5 × 6 = 30 and 5 + 6 = 11

Step 3: Split the middle term

2×2+5x+6x+152x² + 5x + 6x + 152x2+5x+6x+15

Step 4: Factor by grouping

(2×2+5x)+(6x+15)(2x² + 5x) + (6x + 15)(2x2+5x)+(6x+15)

Taking out common factors:

$$x(2x+5)+3(2x+5)$$

Step 5: Factor out the common binomial factor

$$(2x+5)(x+3)$$

Thus, 2x² + 11x + 15 = (2x + 5)(x + 3)

Example 3: Factorize 6x² – 5x – 6

Step 1: Identify the coefficients

• Here, a = 6, b = -5, c = -6
• The product a × c = 6 × (-6) = -36

Step 2: Find two numbers that multiply to -36 and add to -5

• The numbers -9 and 4 satisfy this condition because -9 × 4 = -36 and -9 + 4 = -5

Step 3: Split the middle term

6×2−9x+4x−66x² – 9x + 4x – 66x2−9x+4x−6

Step 4: Factor by grouping

(6×2−9x)+(4x−6)(6x² – 9x) + (4x – 6)(6x2−9x)+(4x−6)

Taking out common factors:

$$3x(2x-3)+2(2x-3)$$

Step 5: Factor out the common binomial factor

$$(3x+2)(2x-3)$$

Thus, 6x² – 5x – 6 = (3x + 2)(2x – 3)

Conclusion

Factorization by splitting the middle term is a useful method for factorizing quadratic expressions of the form ax² + bx + c. The key steps involve:

1. Identifying a, b, and c
2. Finding two numbers that multiply to a × c and add to b
3. Splitting the middle term
4. Factoring by grouping
5. Extracting the common binomial factor

This method provides a systematic way to break down quadratic expressions into simpler factors, making algebraic manipulations easier.