FIND THE ROOTS OF FACTORED POLYNOMIALS

Find the roots of each factored polynomial. 

Example 1 :

(x + 5)(x – 2)

Solution :

(x + 5)(x - 2)  =  0

So, the roots are -5 and 2. 

Example 2 :

(x + 8)(x + 6)

Solution :

(x + 8)(x + 6)  =  0

So, the roots are -8 and -6.

Example 3 :

x(x – 5)

Solution :

x(x - 5)  =  0

So, the roots are 0 and 5.

Example 4 :

(x - 4)(x - 4)

Solution :

(x - 4)(x - 4)  =  0

Here, both the roots are same, that is 4. 

Example 5 :

(2x - 3)(5x - 7)

Solution :

(2x - 3)(5x - 7)  =  0

So, the roots are 3/2 and 7/5. 

Example 6 :

You can model the arch of a fi replace using the equation

y = -(1/9)(x + 18)(x - 18)

where x and y are measured in inches. The x-axis represents the floor. Find the width of the arch at floor level.

Solution :

Use the x-coordinates of the points where the arch meets the floor to find the width. At floor level, y = 0. So, substitute 0 for y and solve for x.

y = -(1/9)(x + 18)(x - 18)

0 = -(1/9)(x + 18)(x - 18)

(x + 18)(x - 18) = 0

Equating each factor to 0, we get

x + 18 = 0 and x - 18 = 0

x = -18 and x = 18

The width is the distance between the x-coordinates, −18 and 18.

So, the width of the arch at floor level is ∣−18 − 18∣= 36 inches.

Example 7 :

You can model the arch of a fi replace using the equation

y = -(1/2)(x + 4)(x - 4)

where x and y are measured in inches. The x-axis represents the floor. Find the width of the arch at floor level.

Solution :

Use the x-coordinates of the points where the arch meets the floor to find the width. At floor level, y = 0. So, substitute 0 for y and solve for x.

y = -(1/2)(x + 4)(x - 4)

0 = -(1/2)(x + 4)(x - 4)

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

Equating each factor to 0, we get

x + 4 = 0 and x - 4 = 0

x = -4 and x = 4

The width is the distance between the x-coordinates, -4 and 4.

So, the width of the arch at floor level is ∣−4 − 4∣ = 8 inches.

Example 8 :

A penguin leaps out of the water while swimming. This action is called porpoising. The height y (in feet) of a porpoising penguin can be modeled by

y = −16x² + 4.8x

where x is the time (in seconds) since the penguin leaped out of the water. Find the roots of the equation when y = 0. Explain what the roots mean in this situation

Solution :

When y = 0

0 = −16x² + 4.8x

Factoring x, we get

-x(16x - 4.8) = 0

Equating each factor to 0, we get

x = 0 and 16x - 4.8 = 0

16x = 4.8

x = 4.8/16

x = 0.3

In 0.3 seconds the penguin lapped out of the water.

Example 9 :

Find the values of x in terms of y that are solutions of each equation.

a. (x + y)(2x − y) = 0

b. (x² − y²)(4x + 16y) = 0

Solution :

a. (x + y)(2x − y) = 0

Equating each factor to 0, we get

x + y = 0 and 2x - y = 0

x = -y and 2x = y

x = -y and x = y/2

b. (x² − y²)(4x + 16y) = 0

(x + y)(x - y) (4x + 16y) = 0

Equating each factor to 0, we get

x + y = 0, x - y = 0 and 4x + 16y = 0

So, the roots are -y, y and -4y.

Example 10 :

Solve (4x − 5 − 16)(3x − 81) = 0

Solution :

(4x − 5 − 16)(3x − 81) = 0

4x - 5 - 16 will become 4x − 21, then 

(4x - 21)(3x - 81) = 0

Equating each factor to 0, we get

So, the roots are 21/4 and 27.

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