Topics to be learn : Factors of a Quadratic Trinomial Factors of a3 + b3 Factors of a3 - b3 Rational Algebraic Expressions

Recall :

Factorising a binomial :

ax + ay = a(x + y)

a² - b² = (a + b)(a – b)

Example,

(1) 4xy + 8xy² = 4xy(1 + 2y)

(2) p² - 9q² = (p)² - (3q)² = (p + 3q)(p - 3q)

Factors of a Quadratic Trinomial :

The general form of a quadratic trinomial is ax² + bx + c.

(x + a)(x + b) = x² + (a + b)x + ab

(1) To find the factors of x² + 12x + 20, compare it with x² + (a + b)x + ab,

We get, a + b = 12 and ab = 2 x 10 = 20.

10 + 2 = 12 and 10 x 2 = 20.

∴ x² + 12x + 20 = x² + (10 + 2)x + 10 × 2

= x² + 10x + 2x + 10 × 2

= x(x + 10) + 2(x + 10)

= (x + 10)(x + 2)

Examples :

(1) Factorise : 2x² - 9x + 9.

Solution:

First we find the product of the coefficient of the square term and the constant term. Here the product is 2 × 9 = 18.

Now, find factors of 18 whose sum is -9, that is equal to the coefficient of the middle term.

18 = (-6) × (-3) ; (-6) + (-3) = -9

Write the term -9x as -6x - 3x

2x² - 9x + 9

= 2x² - 6x - 3x + 9

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

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

(2) Find the factors of 2y² - 4y - 30.

Solution :

2y² - 4y - 30

= 2(y² - 2y - 15)                   ........ taking out the common factor 2

= 2(y² - 5y + 3y - 15)           ........ [-15 = -5 x 3; (-5) + (3) = -2]

= 2 [y(y - 5) + 3(y - 5)]

= 2(y - 5)(y + 3)

Factors of a³ + b³ :

We know that, (a + b)³ = a³ + 3a²b + 3ab² + b³, which we can write as

(a + b)³ = a³ + b³ + 3ab(a + b)

Now, a³ + b³ + 3ab(a + b) = (a + b) ³     .......... interchanging the sides.

∴ a³ + b³ = (a + b) ³ - 3ab(a + b)

= [(a + b)(a + b)²] - 3ab(a + b)

= (a + b)[(a + b)² - 3ab] = (a + b) (a² + 2ab + b² - 3ab)

= (a + b) (a² - ab + b²)

∴ a³ + b³ = (a + b) (a² - ab + b²)

Examples :

(1) x³ + 27y³

= x³ + (3y) ³

= (x + 3y) [x² - x(3y) + (3y)²]

= (x + 3y) [x² - 3xy +9y²]

(2) 8p³ + 125q³

= (2p) ³ + (5q) ³

= (2p + 5q) [(2p)² - 2p × 5q + (5q)²]

= (2p + 5q) (4p² - 10pq + 25q²)

 Factors of a³ - b³ :

(a - b)³ = a³ - 3a²b + 3ab² - b³ = a³ - b³ -3ab(a - b)

Now, a³ - b³ - 3ab (a - b) = (a - b) ³

∴ a³ - b³ = (a - b)³ + 3ab (a - b)

= [(a - b)(a - b)² + 3ab (a - b)]

= (a - b) [(a - b)² + 3ab]

= (a - b) (a² - 2ab + b² + 3ab)

= (a - b) (a² + ab + b²)

∴ a³ - b³ = (a - b)(a² + ab + ^b2)

Examples :

(1) x³ - 8y³

= x³ - (2y)³

∴ x³ - 8y³ = x³ - (2y)³

= (x - 2y) (x² + 2xy + 4y²)

(2) Simplify : (2x + 3y)³ - (2x - 3y)³

Solution :

Using the formula a3 - b3 = (a - b) (a2 + ab + b2)

∴ (2x + 3y)³ - (2x - 3y)³

= [(2x + 3y) - (2x - 3y)] [(2x + 3y)² + (2x + 3y) (2x - 3y) + (2x - 3y)²]

= [2x + 3y - 2x + 3y] [4x² + 12xy + 9y² + 4x² - 9y² + 4x² - 12xy + 9y²]

= 6y(12x² + 9y²) = 72x²y + 54y³

Rational algebraic expressions :

If A and B are two algebraic expressions, B ≠ 0, then \(\frac{A}{B}\) is called a rational algebraic expression.

To simplify a rational algebraic expression, follow the operations as performed on rational numbers.

Examples :

(1) Simplify : \(\frac{a^2+5a+6}{a^2-a-12}\)×\(\frac{a-4}{a^2-4}\)

Solution :

\(\frac{a^2+5a+6}{a^2-a-12}\)×\(\frac{a-4}{a^2-4}\)

= \(\frac{(a+3)(a+2)}{(a-4)(a+3)}\)×\(\frac{a-4}{a^2-4}\)

= \(\frac{1}{a-2}\)

(2) Simplify : \(\frac{a^2-9b^2}{a^3-27b^3}\)

Solution :

\(\frac{a^2-9b^2}{a^3-27b^3}\)

= \(\frac{(a+3b)(a-3b)}{(a-3b)(a^2+3ab+9b^2)}\)

= \(\frac{a+3b}{a^2+3ab+9b^2}\)