FINDING ZEROS OF POLYNOMIALS

Find the zeros of :

Problem 1 :

2x² - 5x - 12

Solution :

2x² - 5x - 12 = 0

2x² - 8x + 3x - 12 = 0

2x(x - 4) + 3(x - 4) = 0

(2x + 3) (x - 4) = 0

2x + 3 = 0 and x - 4 = 0

2x + 3 = 0

2x = -3

x = -3/2

x - 4 = 0

x = 4

So, the zeros of the polynomials is 4, -3/2.

Problem 2 :

x² + 6x + 10

Solution :

x² + 6x + 10 = 0

Using quadratic formula.

a = 1, b = 6, c = 10

b² - 4ac = 6² - 4(1)(10)

= 36 - 40

= -4

So, the zeros of the polynomials is -3 ± i.

Problem 3 :

x² - 6x + 6

Solution :

x² - 6x + 6 = 0

Using quadratic formula.

a = 1, b = -6, c = 6

b² - 4ac = (-6)² - 4(1)(6)

= 36 - 24

= 12

So, the zeros of the polynomials is 3 ± √3.

Problem 4 :

x³ - 4x

Solution :

x³ - 4x = 0

x(x² - 4) = 0

x = 0 and (x² - 4) = 0

x² = 4

Squaring on each sides.

x = ±2

So, the zeros of the polynomials is 0, ± 2.

Problem 5 :

x³ + 2x

Solution :

x³ + 2x = 0

x(x² + 2) = 0

x = 0 and (x² + 2) = 0

x² = -2

Squaring on each sides.

x = ±i√2

So, the zeros of the polynomials is 0, ± i√2.

Problem 6 :

x⁴ + 4x² - 5

Solution :

x⁴ + 4x² - 5 = 0

x⁴ - x² + 5x² - 5 = 0

x²(x² - 1) + 5(x² - 1) = 0

(x² + 5) (x² - 1) = 0

x² + 5 = 0 and x² - 1 = 0

x² + 5 = 0

x² = -5

Squaring on each sides.

x = ±i√5

x² - 1 = 0

x² = 1 

Squaring on each sides.

x = ±√1

So, the zeros of the polynomials is ±√1, ± i√5.

Examples of Finding Roots

Find the roots of :

Problem 7 :

5x² = 3x + 2

Solution :

5x² = 3x + 2

5x² - 3x - 2 = 0

5x² - 5x + 2x - 2 = 0

5x(x - 1) + 2(x - 1) = 0

(5x + 2) (x - 1) = 0

5x + 2 = 0

5x = -2

x = -2/5

x - 1 = 0

x = 1

So, the roots are the polynomials is 1, -2/5.

Problem 8 :

(2x + 1) (x² + 3) = 0

Solution :

(2x + 1) (x² + 3) = 0

2x + 1 = 0 and x² + 3 = 0

2x + 1 = 0

2x = -1 

x = -1/2

x² + 3 = 0

x² = -3

Squaring on each sides.

x = ±i√3

So, the roots are the polynomials is -1/2,  ±i√3.

Problem 9 :

-2z(z² - 2z + 2) = 0

Solution :

-2z(z² - 2z + 2) = 0

-2z = 0 

z = 0

z² - 2z + 2 = 0

a = 1, b = -2 and c = 2

Using quadratic formula.

b² - 4ac = (-2)² - 4(1)(2)

= 4 - 8

= -4

So, the roots are the polynomials is 0, 1 ± i.

Problem 10 :

x³ = 5x

Solution :

x³ - 5x = 0

x(x² - 5) = 0

x = 0 and x² - 5 = 0

x² = 5

Squaring on each sides.

x = ±√5

So, the roots are the polynomials is 0, ±√5 .

Problem 11 :

x³ + 5x = 0

Solution :

x³ + 5x = 0

x(x² + 5) = 0

x = 0 and x² + 5 = 0

x² + 5 = 0

x² = -5

Squaring on each sides.

x = ±i√5

So, the roots are the polynomials is 0, ±i√5 .

Problem 12 :

x⁴ = 3x² + 10

Solution :

x⁴ = 3x² + 10

x⁴ - 3x² - 10 = 0

x⁴ - 5x² + 2x² - 10 = 0

x²(x² - 5) + 2(x² - 5) = 0

(x² + 2) (x² - 5) = 0

x² + 2 = 0 and x² - 5 = 0

x² + 2 = 0

x² = -2

Squaring on each sides.

x = ±i√2

x² - 5 = 0 

x² = 5

Squaring on each sides.

x = ±√5

So, the roots are polynomials is ±i√2, ±√5