Verified answer

Note that:

(3x^3 + 22x^2 + 49x + 14)/(3x + 1)

= (3x^3 + 22x^2 + 49x + 14)/[3(x + 1/3)]

= 1/3 * (3x^3 + 22x^2 + 49x + 14)/(x + 1/3).

Computing (3x^3 + 22x^2 + 49x + 14)/(x + 1/3) with synthetic division:

-1/3. . . . .|. . . . .3. . . . .22. . . . .49. . . . .14

. . . . . . . . . . . . . . . . . . -1 . . . . .-7 . . . . .-14

. . . . . . . ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

. . . . . . . . . . . . .3. . .. .21 . . . . .42. . . . . .0

This gives (3x^3 + 22x^2 + 49x + 14)/(x + 1/3) = 3x^2 + 21x + 42.

Thus:

(3x^3 + 22x^2 + 49x + 14)/(3x + 1)

= 1/3 * (3x^3 + 22x^2 + 49x + 14)/(x + 1/3)

= 1/3 * (3x^2 + 21x + 42)

= x^2 + 7x + 14.

I hope this helps!

... x^3 - 3x - 2 look for a promptly answer to: 0 = x^3 - 3x - 2 ? 2 works (8 - 6 - 2) as a result one ingredient could desire to be (x-2). you could now do your synthetic branch. usually, you may desire to look for values which comprise ±a million, ±2, ±3, ±a million/2, and so on. If it is not obtrusive, start up homing in on a quantity via using estimates on your calculator.

(3x+1) means x= -1/3

all exponential values are present

3,22,49,14

bring down the

3

multiply by -1/3

add to 22

21

multiply by -1/3

add to 49

42

multiply by -1/3

add to 14

0

(3x+1) is a factor

bring down the coefficients

3, 21, 42

plug in the exponents

3x^2+21+42

3(x^2+7x+6)

3(x+6)(x+1)

all together

3(3x+1)(x+6)(x+1)

x= -1/3, -6, -1

You can also use a bit of algebra to do this:

= 3x³ + 22x² + 49x + 14

= x²(3x+1) + 21x² + 49x + 14

= x²(3x+1) + 7x(3x+1) + 42x + 14

= x²(3x+1) + 7x(3x+1) + 14(3x + 1)

Now divide through by (3x+1) to get

x² + 7x + 14

I'm pretty sure you can't and you just have to use the long division method. I'm not positive though.