What is the result when x^3+ 7x^2+ 9x - 8 is divided by x+ 2?

So the quotient is and the remainder is .

Step-by-step explanation:

We could do this by synthetic division since the denominator is a linear factor in the form .

Since we are dividing by , this is our setup for the synthetic division:

-2 |     1    7    9    -8

   |         -2   -10     2

   ______________

          1   5    -1       -6

So the quotient is and the remainder is .

So .

We can also do long division.

                 x^2+5x-1

               ____________________

        x+2| x^3+7x^2+9x-8

              -(x^3+2x^2)

              -------------------

                       5x^2+9x-8

                    -( 5x^2+10x)

                      --------------------

                                 -x-8

                               -(-x-2)

                                --------------

                                      -6

So we see here we get the same quotient, . and the same remainder, .

Now let's check our result that:

.

So I'm going to rewrite the right hand side as a single fraction:

.

Now let's focus on multiplying .

We are going to multiply the first term of the first ( ) to every term in the second ( ).

We are also going to multiply the second term of the first ( ) to every term in the second ( ).

---------------------------Combine like terms:

So let's go back where we were in our check of :

We have the exact same thing on both sides so we did good.

The primes are 2, 3, 5, 7, 11, 13, 17, 19. There are 8 of them, so they make up 40% of the numbers 1–20.

Step-by-step explanation:

We are given to find the percentage of the first 20 natural numbers that are prime.

We know that  the first 20 natural numbers are 1, 2, 3, 4,  .  .  .  , 19, 20.

And, the prime numbers among them are 2, 3, 5, 7, 11, 13, 17 and 19.

So, if S be the sample space for the experiment and A be the event of selecting a prime number.

Then, we have  n(S) = 20  and  n(A) = 8.

Therefore, the probability of event A is given by

P(A) = n(A) / n(S) = 8/20 = 2/5 x 100% = 40%

Thus, the required percentage is 40%.