**Factorization of Trinomial**

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__Factorization of trinomial expressions of the form x__^{2}
+ px + q**: **

__Factorization of trinomial expressions of the form x__

^{2}+ px + qLet’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)

= x^{2} + bx + ax + ab

= x^{2} + (a + b)x + ab

If we compare the product x^{2} +
(a + b)x + ab with x^{2} + px + q, then

p
= a+b and q = ab

So, while factorizing a trinomial
expression of the form x^{2} + 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 x ^{2} + 7x +
12, we have 4 + 3 = 7 and 4 × 3 = 12. So the required values of a = 4 and b = 3.
Now,*

* x ^{2} +
7x + 12*

* = x ^{2}
+ (4 + 3)x + 12*

* = x ^{2}
+ 4x + 3x + 12*

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

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

Note: The
same process is applied for x^{2} – px + q

*For example:** In x ^{2} - 9x +
20, we have 5 + 4 = 9 and 5 × 4 = 20. So the required values of a = 5 and b =
4. Now,*

* x ^{2} -
9x + 20*

* = x ^{2}
- (5 + 4)x + 20*

* = x ^{2}
- 5x - 4x + 20*

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

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

__Factorization of trinomial expressions of the form x__^{2}
+ px - q**: **

__Factorization of trinomial expressions of the form x__

^{2}+ px - qWhile factorizing a trinomial expression
of the form x^{2} + 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 x ^{2} + 3x -
18, we have 6 - 3 = 3 and 6 × 3 = 18. So the required values of a = 6 and b =
3. Now,*

* x ^{2} +
3x - 18*

* = x ^{2}
+ (6 - 3)x - 18*

* = x ^{2}
+ 6x - 3x - 18*

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

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

Note: The
same process is applied for x^{2} – px - q

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

* x ^{2} -
5x - 14*

* = x ^{2}
- (7 - 2)x - 14*

* = x ^{2}
- 7x + 2x - 14*

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

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

__Factorization of trinomial expressions of the form px__^{2}
+ qx + r**:**

__Factorization of trinomial expressions of the form px__

^{2}+ qx + rIn the
trinomial expressions of the form px^{2} + qx + r, p and q are the
numerical coefficients of x^{2} 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 2x ^{2} + 7x
+ 3, we have 6 + 1 = 7 and 6 × 1 = 2 × 3. So the required values of a = 6 and b
= 1. Now,*

* 2x ^{2}
+ 7x + 3*

* = 2x ^{2}
+ (6 + 1)x + 3*

* = 2x ^{2}
+ 6x + x + 3*

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

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

Note: The
same process is applied for px^{2} – qx + r

*For example:** In 9x ^{2} - 12x
+ 4, we have 6 + 6 = 12 and 6 × 6 = 9 × 4. So the required values of a = 6 and
b = 6. Now,*

* 9x ^{2}
- 12x + 4*

* = 9x ^{2}
- (6 + 6)x + 4*

* = 9x ^{2}
- 6x - 6x + 4*

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

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

__Factorization of trinomial expressions of the form px__^{2}
+ qx - r**:**

__Factorization of trinomial expressions of the form px__

^{2}+ qx - rTo factorize
the expressions of the form px^{2} + 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 4x ^{2} + 4x
- 3, we have 6 - 2 = 4 and 6 × 2 = 4 × 3. So the required values of a = 6 and b
= 2. Now,*

* 4x ^{2}
+ 4x - 3*

* = 4x ^{2}
+ (6 - 2)x - 3*

* = 4x ^{2}
+ 6x - 2x - 3*

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

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

Note: The
same process is applied for px^{2} – qx - r

*For example:** In 15x ^{2} - 17x
- 4, we have 20 - 3 = 17 and 20 × 3 = 15 × 4. So the required values of a = 20
and b = 3. Now,*

* 15x ^{2}
- 17x - 4*

* = 15x ^{2}
- (20 - 3)x - 4*

* = 15x ^{2}
- 20x + 3x - 4*

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

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

__Factorization of trinomial expressions of the form a__^{2}
+ 2ab + b^{2}**:**

__Factorization of trinomial expressions of the form a__

^{2}+ 2ab + b^{2}Let’s find the
product of (a + b) and (a + b)

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

= a^{2} + ab + ab
+ b^{2}

= a^{2} + 2ab + b^{2}

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

*For example:** 4x ^{2} + 20x + 25 = (2x)^{2} + 2.2x.5 + 5^{2}*

* = (2x + 5) ^{2}*

*And, 4a ^{2} –
12a + 9 = (2a)^{2} – 2.2a.3 + 3^{2}*

* = (2a – 3) ^{2}*

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

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