X^5-2x^4-24x^3=0. solve by factoring

Answer:

1. x=6

2. x = -4

3. x³ = 0

Step-by-step explanation:

Step 1: Equation at the end of step 1

((x⁵) - ( 2 · (x⁴)))-(2³· 3x³) = 0

Step 2:

Equation at the end of step 2:

((x⁵)-2x⁴)-(2³ · 3x³) = 0

Step 3: X

Step 4: Pulling out like terms

Pull out like factors :

x⁵ - 2x⁴ -24x³ = x³· (x² - 2x -24)

Trying to factor by splitting the middle term

Factoring  x² -2x - 24

The first term is, x² ts coefficient is  1 .

The middle term is,  -2x  its coefficient is  -2 .

The last term, "the constant", is  -24  

Step-1 : Multiply the coefficient of the first term by the constant   1 • -24 = -24  

Step-2 : Find two factors of  -24  whose sum equals the coefficient of the middle term, which is   -2 .

     -24    +    1    =    -23  

     -12    +    2    =    -10  

     -8    +    3    =    -5  

     -6    +    4    =    -2    That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -6  and  4

 x2 - 6x + 4x - 24

Step-4 : Add up the first 2 terms, pulling out like factors :

                   x • (x-6)

             Add up the last 2 terms, pulling out common factors :

                   4 • (x-6)

Step-5 : Add up the four terms of step 4 :

                   (x+4)  •  (x-6)

            Which is the desired factorization

Equation at the end of step

4

:

 x3 • (x + 4) • (x - 6)  = 0  

STEP

5

:

Theory - Roots of a product

5.1    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:

5.2      Solve  :    x3 = 0  

 Solution is  x3 = 0

Solving a Single Variable Equation:

5.3      Solve  :    x+4 = 0  

Subtract  4  from both sides of the equation :  

                     x = -4

Solving a Single Variable Equation:

5.4      Solve  :    x-6 = 0  

Add  6  to both sides of the equation :  

                     x = 6

Supplement : Solving Quadratic Equation Directly

Solving    x2-2x-24  = 0   directly  

Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula

Parabola, Finding the Vertex:

6.1      Find the Vertex of   y = x2-2x-24

Parabolas have a highest or a lowest point called the Vertex .   Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) .   We know this even before plotting  "y"  because the coefficient of the first term, 1 , is positive (greater than zero).  

Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two  x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.  

Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex.  

For any parabola,Ax2+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is   1.0000  

Plugging into the parabola formula   1.0000  for  x  we can calculate the  y -coordinate :  

 y = 1.0 * 1.00 * 1.00 - 2.0 * 1.00 - 24.0

or   y = -25.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = x2-2x-24

Axis of Symmetry (dashed)  {x}={ 1.00}  

Vertex at  {x,y} = { 1.00,-25.00}  

x -Intercepts (Roots) :

Root 1 at  {x,y} = {-4.00, 0.00}  

Root 2 at  {x,y} = { 6.00, 0.00}  

Solve Quadratic Equation by Completing The Square

6.2     Solving   x2-2x-24 = 0 by Completing The Square .

Add  24  to both side of the equation :

  x2-2x = 24

Now the clever bit: Take the coefficient of  x , which is  2 , divide by two, giving  1 , and finally square it giving  1  

Add  1  to both sides of the equation :

 On the right hand side we have :

  24  +  1    or,  (24/1)+(1/1)  

 The common denominator of the two fractions is  1   Adding  (24/1)+(1/1)  gives  25/1  

 So adding to both sides we finally get :

  x2-2x+1 = 25

Adding  1  has completed the left hand side into a perfect square :

  x2-2x+1  =

  (x-1) • (x-1)  =

 (x-1)2

Things which are equal to the same thing are also equal to one another. Since

  x2-2x+1 = 25 and

  x2-2x+1 = (x-1)2

then, according to the law of transitivity,

  (x-1)2 = 25

We'll refer to this Equation as  Eq. #6.2.1  

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of

  (x-1)2   is

  (x-1)2/2 =

 (x-1)1 =

  x-1

Now, applying the Square Root Principle to  Eq. #6.2.1  we get:

  x-1 = √ 25

Add  1  to both sides to obtain:

  x = 1 + √ 25

Three solutions were found :

x = 6

x = -4

x3 = 0

What is the slope of this line?
a. -1/3

b. 1/3

c. 2/3

d. -2/3

5x+4=5x+4 is it one solution, no solution, or infinite solutions?

Identify the type of function represented by f(x)= (4).
A. Quadratic
B. Exponential growth
C. Linear
D. Exponential decay

Is the table a linear function?
Yes or no

please solve for me ​

how to use union in the real world?​

I also need to put why

30 POINTS IF U ANSWER!
Alonso paid for repairs on his car, and 3
5
of the bill was for labor costs. How much was the total bill if the cost of the labor was $79.50? Let b = the amount of the total bill.

Which equation and solution is correct?
Five-thirds b = 79.50, and the total bill was $47.70.
Three-fifths (79.50) = b, and the total bill was $127.20.
Five-thirds b = 79.50, and the total bill was $212.00.
Three-fifths (b) = 79.50, and the total bill was $132.50.

If m∠ P = (19x -5) ∘ and Angle P is a right angle, find the value of x;

22. Write the word sentence as an equation. The product of negative 2 and 5 plus a number d = 14.


also please solve and "product" means multiply.​

How are rates and ratios related?

1)the fish vendor sells 5/7 kilos of bangus for 75.50. if you will buy 2 1/7 kilos of bangus,how much will cost?​

Each letter of L I T T R E L L is written on a card. The cards are placed in a basket. Find each probability if Elsa selects a card, leaves it out, and then selects another card.

7. Is this an independent or dependent event? Explain: ______________________________ _____________________________________ _____________________________________

8. What is the probability of selecting a T and then an E?

9. What is the probability of selecting a T, and then another T?

10. P(E, then I)?

Can you use mechanical pencils on the PSAT?

I need help with find this problem out

PLZZZ HELP X IS 44 .............

For each pair of shapes, decide whether or not Shape A is congruent to Shape B. Englan your meaning
First pair

.Write 1.2 - 10-5 in standard form​

Scooter rental company A charges an initial fee of $3 plus $1.15 for every hour. Scooter rental company B charges $1.75 for every hour. A customer finds that after h hours, the scooter rental companies will cost the same.

Which equation represents this situation?


​1.75h+1.15h=3​

​1.75h=3−1.15h​

​1.15h=3+1.75h​

​1.15h+3=1.75h

Put the following equation of a one into a slope -intercept form, simplifying all fractions.

8x+6y=-54

Un laboratorio desea saber la efectividad de un nuevo medicamento para el control de la diabetes. Al realizar un experimento en 10000 individuos encuentra que en el 90 % de ellos el medicamento resulta ser efectivo. Suponiendo que los 10000 individuos son representativos de la muestra, conteste las siguientes preguntas:
a) ¿Cuál es la población?
b) ¿Cuál es la muestra?
c) Identifique el parámetro de interés
d) Identifique la estadística e indique cuál es su valor e) ¿Se conoce el valor del parámetro?

PORFAVOR LA EXPLICACION AL RESPONDER LAS PREGUNTAS

PLEASE HELP WILL MAKE BRAINLIEST!
The expression ____ means 15 minus the product of 5 times the difference of p minus 6. If p = 8, then the value of the expression is 5.FILL IN THE BLANK
A.15 - 5p - 6
B.15 x 5 - (p - 6)
C.(15 - 5) x (p - 6)
D.15 - 5(p - 6)

! ! ! Angles ! ! !
(worth ten points)
~
Which of these statements are true?
A.) ΔA is similar to ΔB.
B.) ΔB is similar to ΔC.
C.) ΔA is similar to ΔC.
D.) All three triangles are similar.

PLEASE ANSWER THIS FAST!!!!!!!!!

If you paint all the walls in a room, how many square feet do you need to cover.
The east wall is 3 yds long
The South wall is 10 ft long but has a window that is a 5x3
The west wall is 3 yds long but has a door, 7x3
The north wall has a closet 6.5 ft wide and a small was that is 0.5 feet wide and extends 2 feet into the room.

PLEASE HELP

Solve for n.

n + 1 = 4(n – 8)
i will give a brainliest

somebody please just answer.......................

It takes 5 minutes for 7 people to drink 25 pints of water.At this rate,how long would it take 15 people to drink 160 pints of water

what does 2 - 2 equal

pls help im really bad at math no links pls​

Which fraction equals a repeating decimal 3/10 6/12 1/3 3/4