Consider the product of the two linear expressions (y+a) and (y+b).

(y+a)(y+b) = y(y+b) + a(y+b) = y² + by + ay + ab = y² + y(a+b) + ab

We can write it as

y² + y(a+b) + ab = (y+a)(y+b) .......(i)

Similarly, Consider the product of the two linear expressions (ay+b) and (cy+d).

(ay+b)(cy+d) = ay(cy+d) + b(cy+d) = acy² + ady + bcy + bd = acy² + y(ad+bc) + bd

We can write it as

acy² + y(ad+bc) + bd = (ay+b)(cy+d) .......(ii)

Equation (i) is Simple Quadratic Polynomial expressed as Product of Two linear Factors

and Equation (ii) is General Quadratic Polynomial expressed as Product of Two linear Factors

Observing the two Formulas, leads us to the method of Factorization of Quadratic Expressions.

In Equation (i),

the product of coefficient of y² and the constant term = ab

and the coefficient of y = a+b = sum of the factors of ab

Similarly, In Equation (ii),

the product of coefficient of y² and the constant term = (ac)(bd) = (ad)(bc)

and the coefficient of y = (ad+bc) = sum of the factors of acbd

So, if we can resolve the product of y² and the constant term into product of two factors in such a way that their sum is equal to the coefficient of y, then we can factorize the quadratic expression.

We discuss the steps involved in the method and apply it to solve a number of problems.

Method of Factoring Trinomials (Quadratics) :

Step 1 :

Multiply the coefficient of y² by the constant term.

Step 2 :

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

Step 3 :

Rewrite the y term as the sum of two terms with these factors as coefficients.

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 Solved Examples.

The examples are so chosen that all the models are covered.

Example 1 :

Factorize 9y² + 26y + 16

Solution :

Let P = 9y² + 26y + 16

Now, follow the five steps listed above.

Step 1:

(Coefficient of y²) 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)

Step 3:

P = 9y² + 26y + 16 = 9y² + 8y + 18y + 16

Step 4:

P = y(9y + 8) + 2(9y + 8)

Step 5:

P = (9y + 8)(y + 2)

Thus, 9y² + 26y + 16 = (9y + 8)(y + 2) Ans.

Example 2 :

Factorize y² + 7y - 78

Solution :

Let P = y² + 7y - 78

Now, follow the five steps listed above.

Step 1:

(Coefficient of y²) x (constant term) = 1 x -78 = -78

Step 2:

We have to express -78 as two factors whose sum = coefficient of y = 7 ;

-78 = -2 x 39 = -2 x 3 x 13 = -6 x 13; (-6 + 13 = 7)

Step 3:

P = y² + 7y - 78 = y² - 6y + 13y - 78

Step 4:

P = y(y - 6) + 13(y - 6)

Step 5:

P = (y - 6)(y + 13)

Thus, y² + 7y - 78 = (y - 6)(y + 13) Ans.

Example 3 :

Factorize 4y² - 5y + 1

Solution :

Let P = 4y² - 5y + 1

Now, follow the five steps listed above.

Step 1:

(Coefficient of y²) x (constant term) = 4 x 1 = 4

Step 2:

We have to express 4 as two factors whose sum = coefficient of y = -5 ;

4 = 4 x 1 = -4 x -1; [(-4) + (-1) = -5]

Step 3:

P = 4y² - 5y + 1 = 4y² - 4y - y + 1

Step 4:

P = 4y(y - 1) - 1(y - 1)

Step 5:

P = (y - 1)(4y - 1)

Thus, 4y² - 5y + 1 = (y - 1)(4y - 1) Ans.

Example 4 :

Factorize 3y² - 17y - 20

Solution :

Let P = 3y² - 17y - 20

Now, follow the five steps listed above.

Step 1:

Coefficient of y² x constant term = 3 x -20 = -60

Step 2:

We have to express -60 as two factors whose sum = coefficient of x = -17 ;

-60 = -20 x 3; (-20 + 3 = -17)

Step 3:

P = 3y² - 17y - 20 = 3y² - 20y + 3y - 20

Step 4:

P = y(3y - 20) + 1(3y - 20)

Step 5:

P = (3y - 20)(y + 1)

Thus, 3y² - 17y - 20 = (3y - 20)(y + 1) Ans.

Example 5 :

Factorize 2 - 5y - 18y²

Solution :

Let P = 2 - 5y - 18y² = -18y² - 5y + 2

Now, follow the five steps listed above.

Step 1:

(Coefficient of y²) x (constant term) = -18 x 2 = -36

Step 2:

We have to express -36 as two factors whose sum = coefficient of y = -5 ;

-36 = -2 x 18 = -2 x 2 x 9 = 4 x -9; [4 + (-9) = -5]

Step 3:

P = -18y² - 5y + 2 = -18y² + 4y - 9y + 2

Step 4:

P = 2y(-9y + 2) + 1(-9y + 2)

Step 5:

P = (-9y + 2)(2y + 1)

Thus, 2 - 5y - 18y² = (-9y + 2)(2y + 1) Ans.

Example 6 :

Factorize (y² + y)² -18(y² + y) + 72

Solution :

Let P = (y² + y)² -18(y² + y) + 72

Put (y² + y) = t; Then P = t² -18t + 72

Now, follow the five steps listed above.

Step 1:

(Coefficient of t²) x (constant term) = 1 x 72 = 72

Step 2:

We have to express 72 as two factors whose sum = coefficient of t = -18 ;

72 = 12 x 6 = -12 x -6; [(-12) + (-6) = -18]

Step 3:

P = t² -18t + 72 = t² - 12t - 6t + 72

Step 4:

P = t(t - 12) - 6(t - 12)

Step 5:

P = (t - 12)(t - 6)

But t = (y² + y);

So, P = (t - 12)(t - 6) = (y² + y - 12)(y² + y - 6)

In each of these two brackets, there is a Quadratic Polynomial which can be factorised using the five steps above.

y² + y - 12 = y² + 4y - 3y - 12 = y(y + 4) - 3(y + 4) = (y + 4)(y - 3)

y² + y - 6 = y² + 3y - 2y - 6 = y(y + 3) - 2(y + 3) = (y + 3)(y - 2)

See how these two Quadratic Polynomials are factorised with the knowledge of the 5 steps.

You might have mastered the 5 steps of factorisation by this time, to write directly like this.

Thus,

P = (y² + y)² -18(y² + y) + 72

= (y² + y - 12)(y² + y - 6)

= (y + 4)(y - 3)(y + 3)(y - 2) Ans.

For more, on Factoring Quadratics, go to

http://www.math-help-ace.com/Factoring-Trinomials.html