Question

find the zeroes of the polynomial 2x³+3x²-18x-27​

Answer 1

Answer:

Zeroes of the given equation are : - 3/2 , 3 , -3

Step-by-step explanation:

Step by step solution :

STEP

1

:

Equation at the end of step 1

 (((2 • (x3)) +  3x2) -  18x) -  27  = 0  

STEP  

2

:

Equation at the end of step

2

:

 ((2x3 +  3x2) -  18x) -  27  = 0  

STEP

3

:

Checking for a perfect cube

3.1    2x3+3x2-18x-27  is not a perfect cube

Trying to factor by pulling out :

3.2      Factoring:  2x3+3x2-18x-27  

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1:  -18x-27  

Group 2:  2x3+3x2  

Pull out from each group separately :

Group 1:   (2x+3) • (-9)

Group 2:   (2x+3) • (x2)

              -------------------

Add up the two groups :

              (2x+3)  •  (x2-9)  

Which is the desired factorization

Trying to factor as a Difference of Squares:

3.3      Factoring:  x2-9  

Theory : A difference of two perfect squares,  A2 - B2  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =

        A2 - AB + BA - B2 =

        A2 - AB + AB - B2 =

        A2 - B2

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 9 is the square of 3

Check :  x2  is the square of  x1  

Factorization is :       (x + 3)  •  (x - 3)  

Equation at the end of step

3

:

 (x + 3) • (x - 3) • (2x + 3)  = 0  

STEP

4

:

Theory - Roots of a product

4.1    A product of several terms equals zero.  

When a product of two or more terms equals zero, then at least one of the terms must be zero.  

We shall now solve each term = 0 separately  

In other words, we are going to solve as many equations as there are terms in the product  

Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation:

4.2      Solve  :    x+3 = 0  

Subtract  3  from both sides of the equation :  

                     x = -3

Solving a Single Variable Equation:

4.3      Solve  :    x-3 = 0  

Add  3  to both sides of the equation :  

                     x = 3

Solving a Single Variable Equation:

4.4      Solve  :    2x+3 = 0  

Subtract  3  from both sides of the equation :  

                     2x = -3

Divide both sides of the equation by 2:

                    x = -3/2 = -1.500

Three solutions were found :

x = -3/2 = -1.500

x = 3

x = -3