# Factorization of Trinomial

## Factorization of Trinomial

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### Factorization of trinomial expressions of the form x2 + px + q:

Let’s consider any two binomial expressions (x + a) and (x + b).

The product of (x + a) and (x + b) = (x + a)(x + b)

= x(x + b) + a(x + b)

= x2 + bx + ax + ab

= x2 + (a + b)x + ab

If we compare the product x2 + (a + b)x + ab with x2 + px + q, then

p = a+b and q = ab

So, while factorizing a trinomial expression of the form x2 + px + q, we should find the two value a and b such that a + b = p and a × b = q. Then, the expression is factorized by grouping.

For example: In x2 + 7x + 12, we have 4 + 3 = 7 and 4 × 3 = 12. So the required values of a = 4 and b = 3. Now,

x2 + 7x + 12

= x2 + (4 + 3)x + 12

= x2 + 4x + 3x + 12

= x(x + 4) + 3(x + 4)

= (x + 4)(x + 3)

Note: The same process is applied for x2 – px + q

For example: In x2 - 9x + 20, we have 5 + 4 = 9 and 5 × 4 = 20. So the required values of a = 5 and b = 4. Now,

x2 - 9x + 20

= x2 - (5 + 4)x + 20

= x2 - 5x - 4x + 20

= x(x - 5) - 4(x - 5)

= (x - 5)(x - 4)

### Factorization of trinomial expressions of the form x2 + px - q:

While factorizing a trinomial expression of the form x2 + px - q, we should find the two value a and b such that a - b = p and a × b = q. Then, the expression is factorized by grouping.

For example: In x2 + 3x - 18, we have 6 - 3 = 3 and 6 × 3 = 18. So the required values of a = 6 and b = 3. Now,

x2 + 3x - 18

= x2 + (6 - 3)x - 18

= x2 + 6x - 3x - 18

= x(x + 6) - 3(x + 6)

= (x + 6)(x - 3)

Note: The same process is applied for x2 – px - q

For example: In x2 - 5x - 14, we have 7 - 2 = 5 and 7 × 2 = 14. So the required values of a = 7 and b = 2. Now,

x2 - 5x - 14

= x2 - (7 - 2)x - 14

= x2 - 7x + 2x - 14

= x(x - 7) + 2(x - 7)

= (x - 7)(x + 2)

### Factorization of trinomial expressions of the form px2 + qx + r:

In the trinomial expressions of the form px2 + qx + r, p and q are the numerical coefficients of x2 and x respectively and r is the constant term.

To factorize such expressions, we need to find the two values a and b such that a + b = q and a × b = p × r. Then, the expression is expanded to four terms and factorization is performed by grouping.

For example: In 2x2 + 7x + 3, we have 6 + 1 = 7 and 6 × 1 = 2 × 3. So the required values of a = 6 and b = 1. Now,

2x2 + 7x + 3

= 2x2 + (6 + 1)x + 3

= 2x2 + 6x + x + 3

= 2x(x + 3) + 1(x + 3)

= (x + 3)(2x + 1)

Note: The same process is applied for px2 – qx + r

For example: In 9x2 - 12x + 4, we have 6 + 6 = 12 and 6 × 6 = 9 × 4. So the required values of a = 6 and b = 6. Now,

9x2 - 12x + 4

= 9x2 - (6 + 6)x + 4

= 9x2 - 6x - 6x + 4

= 3x(3x - 2) - 2(3x - 2)

= (3x - 2)(3x - 2)

### Factorization of trinomial expressions of the form px2 + qx - r:

To factorize the expressions of the form px2 + qx + r, we need to find the two values a and b such that a - b = q and a × b = p × r. Then, the expression is expanded to four terms and factorization is performed by grouping.

For example: In 4x2 + 4x - 3, we have 6 - 2 = 4 and 6 × 2 = 4 × 3. So the required values of a = 6 and b = 2. Now,

4x2 + 4x - 3

= 4x2 + (6 - 2)x - 3

= 4x2 + 6x - 2x - 3

= 2x(2x + 3) - 1(2x + 3)

= (2x + 3)(2x - 1)

Note: The same process is applied for px2 – qx - r

For example: In 15x2 - 17x - 4, we have 20 - 3 = 17 and 20 × 3 = 15 × 4. So the required values of a = 20 and b = 3. Now,

15x2 - 17x - 4

= 15x2 - (20 - 3)x - 4

= 15x2 - 20x + 3x - 4

= 5x(3x - 4) + 1(3x - 4)

= (3x - 4)(5x + 1)

### Factorization of trinomial expressions of the form a2 + 2ab + b2:

Let’s find the product of (a + b) and (a + b)

(a + b)(a + b) = a(a+b)+b(a+b)

= a2 + ab + ab + b2

= a2 + 2ab + b2

Thus, (a + b)2 are the factors of a2 + 2ab + b2. Similarly, (a – b)2 are the factors of a2 – 2ab + b2. So, the expression of the form a2 + 2ab + b2 is factorized by making it a perfect square trinomial.

For example: 4x2 + 20x + 25 = (2x)2 + 2.2x.5 + 52

= (2x + 5)2

And, 4a2 – 12a + 9 = (2a)2 – 2.2a.3 + 32

= (2a – 3)2

You can comment your questions or problems regarding the factorization of trinomial here.