Which of the following is a factor of 6x^3 + 6?
a) x - 1
b) x + 1
c) x² - 2x + 1
d) x² + x + 1

B

Step-by-step explanation:

Given

6x³ + 6 ← factor out 6 from each term

= 6(x³ + 1)

x³ + 1 is a sum of cubes and factors as

x³ + 1 = (x + 1)(x² - x + 1)

Hence

6x³ + 6 = 6(x + 1)(x² - x + 1)

With factor (x + 1) → B

b) x + 1

Step-by-step explanation:

you can either

1) take (6x³ + 6) and divide by all the choices to see which one gives you a factor. You will realize that if you divide this by option b, you will be able to factorize the equation as follows:

(6x³ + 6) = 6(x+1)(x²−x+1)

Hence option b is a factor

or

2) (my preferred method), utilize the properties of functions and roots.

Let function f(x) = 6x³ + 6

any value of a which gives f(a) = 0 is a root , i.e (x-a) is a factor.

In this case, lets consider option b

let x + 1 = 0 > or x = -1

substitute this into the function f(x)

f(-1) = 6 (-1)³ + 6

f(-1) = -6 + 6 = 0

hence x = -1 is a root , or (x+1) is a factor.

as a sanity check, lets try choice a) x -1

let x - 1 = 0 > or x = +1

substitute this into the function f(x)

f(1) = 6 (1)³ + 6

f(1) = 6 + 6 = 12 ≠0

hence x = 1 is NOT a root , or (x-1) is NOT a factor.

You can do the same for c and d and find that they too are NOT factors.

answer: e 9, 5.75

step-by-step explanation:

step 1

find the variation ab in the x coordinate

delta x=(x2-x1)=10-> 8 units

find the variation ab  in the y coordinate 

delta y=(y2-y1)=6-> 2 units

step 2

find the coordinates point c

  x coordinate 

cx=ax+delta x*2/> cx=2+8*(2/> cx=3

  y coordinate 

cy=ay+delta y*2/> cy=4+2*(2/> cy=4.25

the coordinates of point c are (3,4.25)

step 3

find the coordinates point d

  x  coordinate 

dx=ax+delta x*4/> dx=2+8*(4/> dx=4

  y  coordinate 

dy=ay+delta y*4/> dy=4+2*(4/> dy=4.5

the coordinates of point d are (4, 4.5)

step 4

find the coordinates point e

  x  coordinate 

ex=ax+delta x*4/> ex=2+8*(6/> ex=5

  y  coordinate 

ey=ay+delta y*4/> ey=4+2*(6/> ey=4.75

the coordinates of point e are (5, 4.75)

step 5

find the coordinates point f

  x  coordinate 

fx=ax+delta x*4/> fx=2+8*(8/> fx=6

  y  coordinate 

fy=ay+delta y*4/> fy=4+2*(8/> fy=5

the coordinates of point f are (6,5)

step 6

find the coordinates point g

  x  coordinate 

gx=ax+delta x*4/> gx=2+8*(10/> gx=7

  y  coordinate 

gy=ay+delta y*4/> gy=4+2*(10/> gy=5.25

the coordinates of point g are (7,5.25)

step 7

find the coordinates point h

  x  coordinate 

hx=ax+delta x*4/> hx=2+8*(12/> hx=8

  y  coordinate 

hy=ay+delta y*4/> hy=4+2*(12/> hy=5.5

the coordinates of point h are (8,5.5)

step 8

find the coordinates point i

  x  coordinate 

ix=ax+delta x*4/> ix=2+8*(14/> ix=9

  y  coordinate 

iy=ay+delta y*4/> iy=4+2*(14/> iy=5.75

the coordinates of point i are (9,5.75)

- r3ktforgood ☕

the answer is b