Learn How to Factorize the Trinomial x^{2} + px +q? A Trinomial is a three-term algebraic expression. By Factoring the trinomial expression, we will get the product of two binomial terms. Here, the trinomial expression contains three terms which are combined with the operations like addition or subtraction. Here, we need to find the coefficient values, and based on the coefficient values, we can find out the binomial terms as products of trinomial expression.

## How to Factorize the Trinomial x^{2} + px + q?

To find x^2 + px + q, we have to find the two terms (m + n) = p and mn = q.

Substitute (m + n) = p and mn = q in x^2 + px + q.

x^2 + px + q = x^2 + (m + n)x + mn.

By expanding the above expression, we will get

x^2 + px + q = x^2 + mx + nx + mn.

separate the common terms from the above expression.

that is, x(x + m) + n(x + m).

factor out the common term.

that is, (x + m) (x + n).

So, x^2 + px + q = (x + m)(x + n).

### Factorization of Trinomial Steps

- Note down the given trinomial expression and compare the expression with the basic expression.
- Find out the product and sum of co-efficient values that is (m + n) and mn.
- Based on the above step, find out the two co-efficient values m and n.
- Finally, we will get the product of two terms which are equal to the trinomial expression.

### Examples on Factoring Trinomials of Form x^{2} + px + q

1. Resolve into factors

(i) a^{2} + 3a -28

Solution:

Given Expression is a^{2} + 3a -28.

Compare the a^{2} + 3a -28 with the x^2 + px +q

Here, p = m + n = 3 and q = mn = -28

q is the product of two co-efficient. That is, 7 *(- 4) = -28

p is the sum of two co-efficient. That is 7 + ( – 4) = 3.

So, a^{2} + 3a -28 = a^{2} + [7 + (-4)]a – 28.

= a^{2} + 7a – 4a – 28.

=a (a + 7) – 4(a + 7)

Factor out the common term.

That is, (a + 7) (a – 4).

Finally, the expression a^{2} + 3a -28 = (a + 7) (a – 4).

(ii) a^{2} + 8a + 15

Solution:

Given Expression is a^{2} + 8a + 15.

Compare the a^{2} + 8a + 15 with the x^2 + px +q.

Here, p = m + n = 8 and q = mn = 15.

q is the product of two co-efficient. That is, 5 * 3 = 15.

p is the sum of two co-efficient. That is 5 + 3 = 8.

So, a^{2} + 8a + 15 = a^{2} + (5 + 3)a + 15.

= a^{2} + 5a + 3a + 15.

=a (a + 5) + 3(a + 5).

Factor out the common term.

That is, (a + 5) (a + 3).

Finally, the expression a^{2} + 8a + 15= (a + 5) (a + 3).

2. Factorize the Trinomial

(i) a^{2} + 15a + 56

Solution:

Given Expression is a^{2} + 15a + 56.

Compare the a^{2} + 15a + 56 with the x^2 + px +q.

Here, p = m + n = 15 and q = mn = 56.

q is the product of two co-efficient. That is, 7 * 8 = 56.

p is the sum of two co-efficient. That is 7 + 8 = 15.

So, a^{2} + 15a + 56= a^{2} + (7 + 8) a + 56.

= a^{2} + 7a + 8a + 56.

=a (a + 7) + 8(a + 7).

Factor out the common term.

That is, (a + 7) (a + 8).

Finally, the expression a^{2} + 15a + 56= (a + 7) (a + 8).

(ii) a^{2} + a – 56

Solution:

Given Expression is a^{2} + a – 56.

Compare the a^{2} + a – 56 with the x^2 + px +q.

Here, p = m + n = 1 and q = mn = – 56.

q is the product of two co-efficient. That is, – 7 * 8 = – 56.

p is the sum of two co-efficient. That is – 7 + 8 = 1.

So, a^{2} + a – 56 = a^{2} + ( – 7 + 8)a – 56.

= a^{2} – 7a + 8a – 56.

=a (a – 7) + 8(a – 7).

Factor out the common term.

That is, (a – 7) (a + 8).

Finally, the expression a^2 + a – 56 = (a – 7) (a + 8).