Question

Use synthetic division to show that the number given to the right of the equation is a solution to the equation, then solve for the solution set.

x^3 - 5x^2 + 2x + 8 = 0 ; -1
the number -1 is a solution to the equation because the remainder of the division, x^3 - 5x^2 + 2x + 8 divided by x + 1 is ___.

The solution set is {___}
(Use a comma to separate answers as needed.)

Thank you!!!!

Answer 1

x^3 - 5x^2 + 2x + 8 = 0

-1 is a one solution to the equation because the remainder of the division , x^3 - 5x^2 + 2x + 8 divided by  x + 1

Put  f(x)  = x^3 -5x^2 + 2x  + 8

f(-1) = (-1)^3 - 5(-1)^2 + 2(-1) + 8

        = -1 -5 -2 + 8

       = -8 + 8

f(-1) = 0

There fore x + 1 is a factor of  x^3 -5x^2 + 2x + 8

f(x) =(x + 1)(x^2 -6x  + 8)

But f(x) = 0

There fore (x + 1)(x^2 -6x + 8) = 0

x + 1 =0 0r (x^2 - 6x + 8) = 0

x^2 -6x + 8 = 0

x^2 - 4x - 2x + 8 = 0

Take out common term x and -2

x(x  - 4) -2(x - 4) = 0

Take out common term x - 4

(x - 4)(x  - 2) = 0

x - 4 = 0 or x - 2 = 0

x - 4 = 0

Add 4 to each side

There fore x = 4

x - 2 = 0

Add 2 to each side

x = 2

The solution set is {-1, 2, 4 }.

Comments

Please use mentioned method.Use synthetic division only.

Answer 2

The polynomial function is and the root is - 1.

Use synthatic division to find

Step - 1

Write the terms of the dividend so that the degrees of the terms are in descending order.

Then write just the coefficients as shown at the right.

Step - 2:

Write the constant r of the divisor (x - r) to the left. In this case, r = - 1. Bring the first coefficient, 1, down.

Step - 3:

Multiply the first coefficient by r : - 1*1 = - 1.

Write the product under the second coefficient, - 5 and : - 5 + (- 1) = - 6.

Step - 4:

Multiply the sum, - 6, by r : -1*- 6 =. 6.

Write the product under the next coefficient, 2 and add : 2 + 6 = 8.

Step -  5:

Multiply the sum,8, by r : - 1*8 = 8 .

Write the product under the next coefficient, 8 and add : 8 + (- 8) = 0. The remainder is 0.

The numbers along the bottom row are the coefficients of the quotient. Start with the power of x  that is one less than the degree of the dividend. Thus, the quotient is x² - 6x + 8 = 0.

The resulting equation is x² - 6x + 8 = 0 and solve for x  by using factors method.

x² - 4x - 2x + 8 = 0

x(x - 4) - 2(x - 4) = 0

Take out common term (x - 4).

(x - 4 )(x -2) = 0

Apply zero product property.

x - 4 = 0 and x - 2 = 0

x = 4 and x = 2.

The solution set is { -1, 2, 4}.