1 Write the polynomial as the product of linear factors fx

1) Write the polynomial as the product of linear factors.

f(x) = x² + 81

f(x) =

List all the zeros of the function. (Enter your answers as a comma-separated list.)

x =

2) Write the polynomial as the product of linear factors.

h(x) = x² 6x + 18

h(x) =

List all the zeros of the function. (Enter your answers as a comma-separated list.)

x =

3)Write the polynomial as the product of linear factors.

f(x) = x⁴ 16

f(x) =

List all the zeros of the function. (Enter your answers as a comma-separated list.)

x =

4) Write the polynomial as the product of linear factors.

f(z) = z² 10z + 26

f(z) =

List all the zeros of the function. (Enter your answers as a comma-separated list.)

z =

5) Write the polynomial as the product of linear factors.

h(x) = x³ x + 60

h(x) =

List all the zeros of the function. (Enter your answers as a comma-separated list.)

x =

6) Write the polynomial as the product of linear factors.

f(x) = 5x³ 9x² + 18x + 4

f(x) =

List all the zeros of the function. (Enter your answers as a comma-separated list.)

x =

7) Write the polynomial as the product of linear factors.

g(x) = x⁴ 2x³ + 10x² 18x + 9

g(x) =

List all the zeros of the function. (Enter your answers as a comma-separated list.)

x =

Solution

1) f(x) =x^2 +81

= x^2 - (-9i)^2 ( a^2 - b^2 = (a+b)(a-b) )

= (x -9i)(x +9i)

There are no real zeros only imaginary roots are present:

(x -9i)(x +9i) =0

x = 9i ;x = -9i

2) h(x) = x^2 6x + 18

find the roots of polynomial by quadratic formula:

x = ( -b + /- sqrt(b^2 -4ac) )/2a

= ( 6 + / - sqrt(36 - 4*18) )/2 = ( 6 + / sqrt(-36) )/2

= 3 + /- i*3

= 3 +/ - 3i

So, h(x) = ( x-3 -3i)(x -3 +3i)

zeros are x = 3 +3i , 3-3i

3) f(x) = x4 16

Use the formula : (a^2 -b^2) = (a+b)(a-b)

x^4 -16 = (x^2 +4)(x^2 -4)

= (x +2i)(x-2i)(x+2)( x-2)

f(x) = (x +2i)(x-2i)(x+2)( x-2)

zeros : f(x)=0

(x +2i)(x-2i)(x+2)( x-2) =0

x = -2i , 2i , -2 , 2

4) f(z) = z^2 -10z +26

find the roots of polynomial by quadratic formula:

x = ( -b + /- sqrt(b^2 -4ac) )/2a

z = ( 10 + /- sqrt(100 -4*26) )/2

= ( 10 +/-2i)/2

= 5 +/- i

z = 5 +i , 5 -i

f(z) = (x -5 -i)(x -5 +i)