FACTORIZATION USING ALGEBRAIC IDENTITIES

Factorize each polynomial using algebraic identity.

Problem 1 :

w² - 4

Solution :

w² - 4 = w² - 2²

a² - b² = (a + b) (a - b)

w² - 2² = (w + 2) (w - 2)

Problem 2 :

a² - b²

Solution :

a² - b² = (a + b) (a - b)

Problem 3 :

9 - 16t²

Solution :

9 - 16t² = 3² - (4t)²

a² - b² = (a + b) (a - b)

3² - (4t)² = (3 + 4t) (3 - 4t)

Problem 4 :

8a³ - 27b³

Solution :

8a³ - 27b³ = (2a)³ - (3b)³

a³ - b³ = (a - b) (a² + ab + b²)

= (2a - 3b) [(2a)² + (2a)(3b) + (3b)²]

= (2a - 3b) (4a² + 6ab + 9b²)

Problem 5 :

m³ + n³

Solution :

a³ + b³ = (a + b) (a² - ab + b²)

m³ + n³ = (m + n) (m² - mn + n²)

Problem 6 :

125x³ - 64

Solution :

125x³ - 64 = (5x)³ - 4³

a³ - b³ = (a - b) (a² + ab + b²)

= (5x - 4) [(5x)² + (5x)(4) + (4)²]

= (5x - 4) (25x² + 20x + 16)

Problem 7 :

x² + 10x + 25

Solution :

(a + b)² = a² + 2ab + b²

x² + 10x + 25 = (x)² + 2(x)(5) + (5)²

= (x + 5)²

= (x + 5) (x + 5)

Problem 8 :

36u² - 12uv + v²

Solution :

(a - b)² = a² - 2ab + b²

36u² - 12uv + v² = (6u)² - 2(6u)(v) + (v)²

= (6u - v)²

= (6u - v) (6u - v)

Problem 9 :      

4a² - 4a + 1

Solution :

(a - b)² = a² - 2ab + b²

4a² - 4a + 1 = (2a)² - 2(2a)(1) + 1²

= (2a - 1)²

= (2a - 1) (2a - 1)

Problem 10 :

25x² - 36

Solution :

25x² - 36 = (5x)² - (6)²

a² - b² = (a + b) (a - b)

(5x)² - (6)² = (5x + 6) (5x - 6)

Problem 11 :

If t > 0 and t² − 4 = 0, what is the value of t ?

Solution :

t² − 4 = 0

Using the algebraic identity, 

t² − 2² = 0

(t + 2)(t - 2) = 0

t = -2 and t = 2

Since the condition is t > 0, the value fo t is 2.

Problem 12 :

If x^a²/ x^b² = 16, x > 1 and a + b = 2, what is the value of a - b ?

Solution :

x^a²/ x^b² = 16

x^{a^2-b^2} = 16

Since the bases are equal, we can equate the powers.

a² - b² = 16

(a + b)(a - b) = 16

Applying the value of a + b, we get

2(a - b) = 16

a - b = 16/2

a - b = 8

So, the value fo a - b is 8.

Problem 13 :

x² + y² + 4x - 2y = -1

The equation of a circle in the xy-plane is shown above. What is the radius of the circle ?

a)  2      b)  3     c)  4      d)  9

Solution :

x² + y² + 4x - 2y = -1

x² + 4x + y² - 2y + 1 = 0

x² + 2x(2) + 2² - 2²+ y² - 2y(1) + 1² - 1² + 1 = 0

(x + 2)² + (y - 1)² - 4 - 1 + 1 = 0

(x + 2)² + (y - 1)² = 4

(x + 2)² + (y - 1)² = 2²

Comparing with,

(x - h)² + (y - k)² = r²

Radius = 2

option a is correct.

Problem 14 :

The equation

x² + (y - 1)² = 49

represent circle A. Circle B is obtained by shifting circle A down 2 units in the xy-plane. Which of the following equation represents circle B ?

a)  (x - 2)² + (y - 1)² = 49           b) x² + (y - 3)² = 49

c) (x + 2)² + (y - 1)² = 49           d)  x² + (y + 1)² = 49

Solution :

x² + (y - 1)² = 49

Obtaining circle B by shifting circle A by 2 units down, then 

x² + (y - 1 - 2)² = 49

x² + (y - 3)² = 49

Option b is correct.

Problem 15 :

(d − 30)(d + 30) − 7 = −7

What is a solution to the given equation?

Solution :

(d−30)(d+30) −7 = −7

(d² − 30²) −7 = −7

d² − 900 −7 = −7

d² = −7 + 900 + 7

d² = 900

d = √900

d = -30 and 30