Simple Quadratic Polynomial as Product of Two Factors:

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

Formula 11 in Factoring Special Products :

General Quadratic Polynomial as Product of Two Factors:

acx^{2} + x(ad + bc) + bd = (ax + b)(cx + d)

Observing these two Formulas, we can say that if
the numerical product of the cofficient of x^{2} and constant term
be expressed as two numerical factors such that
their sum is equal to the coefficient of x, then
we can write the factors of the Quadratic Polynomial.

This leads us to a method.

Method of Factoring Trinomials (Quadratics)

Step 1 :

Multiply the coefficient of x^{2} by the constant term.

Step 2 :

Resolve this product into two factors such that
their sum is the coefficient of x

Step 3 :

Rewrite the x term as the sum of
two terms with these coeffiecients.

Step 4 :

Then take the common factor in the
first two terms and the last two terms.

Step 5 :

Then take the common factor from the two terms thus formed.
What you get in step 5 is the product of the required two factors.

The method will be clear by the following Soved Examples.

Example 1 of Factoring Trinomials (Quadratics)

Factorize 9x^{2} + 26x + 16

Solution to Example 1 of Factoring Trinomials (Quadratics) :

Let P = 9x^{2} + 26x + 16

In Factoring of Trinomials (Quadratics) , follow the five steps listed above.

Step 1: Coefficient of x^{2} x constant term
= 9 x 16 = 144

Step 2: We have to express 144 as two factors whose sum
= coefficient of x = 26;
144 = 2 x 72 = 2 x 2 x 36 = 2 x 2 x 2 x 18 = 8 x 18; (8 + 18 = 26)

Thus, Applying the Method of Algebra Factoring of Quadratic Polynomial to
9x^{2} + 26x + 16,
we get the Factors as
(9x + 8)(x + 2) Ans.

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