Use synthetic division

Question

Use synthetic division to show that x is a solution of the third-degree polynomial equation, and use the result to factor the polynomial completely. List all real solutions of the equation. (Enter your answers as a comma-separated list.)

x³ + 6x² + 11x + 6 = 0,    x = −3

Answer 1

Tom got the synthetic division

+3

Just to check the result, or do an alternative way to solve this problem, do long division, if x=-3 is a zero, then x+3 is a factor

(x^3 + 6x^2 + 11x + 6) divided by (x+3) is

similar to arithmetic division

x+3 | x^3 + 6x^2 + 11x + 6

divide x into x^3 to get x^3/x = x^2

multiply x^2 by x+3 to get x^3 +3x^2

subtract x^3 +3x^2 from x^3 +6x^2 to get 3x^2

divide x into 3x^2 to get 3x

multiply 3x by x+3 to get 3x^2 + 9x

subtract 3x^2 + 9x from 3x^2+11x to get 2x

divide x into 2x to get 2

multiply 2 times x+3 to get 2x+ 6

subtract 2x + 6 from 2x + 6 to get zero

leaving the quotient as x^2 + 3x + 2 with no remainder

set x^2 +3x +2 = 0

factor to (x+1)(x+2) = 0

set each factor = 0, x+1= 0, x=-1 and x+2=0, x=-2

or

complete the square

x^2 +3x + 9/4 = -2 + 9/4

(x+3/2)^2 = 1/4

x+3/2 = + or - sqr(1/4) = +1/2 or -1/2

x = 1/2-3/2 or -1/2-3/2 = -1 or -2

or use the quadratic formual

x = -b/2a + or - 1/2a)sqr(b^2 -4ac)

=-3/2 + or - (1/2)sqr(3^2 -4(1)(2))

= -3/2 + or - (1/2)sqr(9-8)

= -3/2 + or -1/2

= -3/2-1/2 or -3/2+1/2

= -2 or -1

the zeros are -1, -2 and -3

Answer 2

-3 | 1 6 11 6

-3 -9 -6

1 3 2 0

The final 0 shows that -3 is a root.

We then factor the polynomial made from the numbers on the bottom row:

x² + 3x + 2 = (x + 2)(x +1)

Roots are: -3, -2, -1