Please help with this question Thank you Solve the polynomia

Please help with this question... Thank you

Solve the polynomial inequality indicated using a number line and the behavior of the graph at each zero. (x + 3)^3 (x - 1)^2 (x - 3) lessthanorequalto 0

Solution

( x+ 3)^3 ( x - 1)^2 ( x- 3) <= 0

the critical points of the inequality are

x = -3

x = 1

x = 3

so dividing the number line into 4 intervals and checking the inequality at each interval

------------------- - 3 ---------------------- 1 --------------------------- 3 -----------------------

1st interval ( -infinity to - 3 )

taking a test point lets say - 4

( -4+ 3)^3 ( -4 - 1)^2 ( -4- 3) <= 0

-1 * 25 *-7 = 175

175 > 0

therefore inequality is false in this interval

2nd interval ( -3 to 1 ) test point 0

( 0+ 3)^3 ( 0 - 1)^2 ( 0- 3) <= 0

27*1*-3

-8 < = 0

hence, inequality is true in this interval

3rd interval ( 1 to 3) test point 2

( 2+ 3)^3 ( 2 - 1)^2 ( 2- 3) <= 0

125*1*-1

-125 < = 0

hence ineqlaity is true

4th interval ( 3 to infinity ) test point 4

( 4+ 3)^3 ( 4 - 1)^2 ( 4- 3) <= 0

343*9*1

3087 > 0

hence its false

so solution region is [ -3 , 3 ]