x^2 + 9x + 14 = (x + 2)(?)?

x^2 + 9x + 14 = (x + 2)(x + 7).

x²  +  9x  + 14   = ( x  + 7 ) ( x  + 2 )

Here is a basic method to divide a polynomial by another. It is long and boring. However, it works, regardless of the degree of the polynomial being divided.

It is an iterative process (meaning = you go through a loop that repeats until you are done)

There are 4 steps in each loop (with the 4th one being simply a recap of what you have so far). After you do a few of these, you can even skip some of the steps.

-----

(x^2 + 9x + 14) / (x + 2)

step 1: Divide the highest-degree term of the numerator (top) by the highest degree of the denominator (below):

x^2 / x = x

step 2: multiply the whole denominator by this result:

x (x + 2) = x^2 + 2x

step 3: subtract this result from the numerator (careful with signs)

(x^2 + 9x + 14) - (x^2 + 2x) = 7x + 14

step 4: recap

(x^2+9x+14)/(x+2) = x , with a remainder of 7x + 14

Repeat the loop, this time using the remainder as the new numerator

(7x + 14) / (x + 2)

step 1:  7x / x = 7

step 2: 7(x + 2) = 7x + 14

step 3: (7x + 14) - (7x + 14) = 0

step 4: recap

(x^2+9x+14)/(x+2) = x + 7 , with a remainder of ZERO !!!

Because our remainder is zero, we stop.

Final answer:

(x^2+9x+14) = (x+2)(x+7)

------

These things will not always divide evenly. The process is stopped when the remainder's degree is less than the degree of the denominator. The remainder is... a remainder.

Easy example

(x^2+9x+15)/(x+2)

step 1: x^2/x = x

step 2: x(x+2) = x^2 + 2x

step 3: (x^2+9x+15) - (x^2 + 2x) = 7x+15

step 4: so far we have x with a remainder of 7x+15

(7x+15)/(x+2)

step 1: 7x/x = 7

step 2: 7(x+2) = 7x+14

step 3: (7x+15) - (7x+14) = 1

step 4: recap

(x^2 + 9x + 15) / (x+2) = (x + 7) with a remainder of 1

final answer

x^2 + 9x + 15 = [(x+2)(x+7)] + 1

Tyyhujnvxcbbhgfdgjjffvbjkjgffgbbn

xx ?? + / ? == ?? (?) + 2

x^2 + 9x + 14 =

x^2 + 7x + 2x + 14 =

x(x + 7) + 2(x + 7) =

(x + 2)(x + 7)

-8x + x^2 + 15 = (x - 5)(x - 3)

x^2 + 9x + 14 = (x + 2)(x + 7) answer//

this is the third question 10 min.

x^2 + 9x + 14 = (x + 2)(x + 7)