Express 3.222......in p/q form
1 answer:
Answer:
3.22222...... =
Step-by-step explanation:
In this question we have to convert the number given in recurrent decimals into fraction.
Recurrent decimal number is 3.22222.......
Let x = 3.2222......... -------(1)
Multiply this expression by 10.
10x = 32.2222........... -------(2)
Now subtract the expression (1) from (2),
10x = 32.22222.....
<u> x = 3.22222.......</u>
9x = 29
x =
Therefore, recurrent decimal number can be written as which is in the form of
.
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Answer:
The quotient when is divided by
is
with no remainder, so x+2 is a factor of p(x).
Step-by-step explanation:
As the question suggests, there are two ways to solve this problem: long division (which can always be used to divide polynomials), and synthetic division (which can only be used when you are dividing by something of the form . In this case, we may use synthetic division, since
is equal to
. I will use synthetic division here as it is slightly faster than long division.
The -2 on the left represents that we are dividing by x-(-2), and the rest of the numbers in the top row are the coefficients of p(x): 1, -4, -8, and 8.
The first step in synthetic division is to being down the leading coefficient of the polynomial: in this case, the 1 (as indicated in red). Now, we multiply the 1 by -2. We get -2, which we place directly below the -4.
Next, we add directly down the column, (-4 + (-2) is equal to -6), and this answer is placed in the box below. We can continue this process, getting the coefficients 1, -6, 4, and 0 in the bottom row.
This is the answer: the quotient is , and the remainder is zero (which indicates that x+2 is a factor of p(x)).
--
Edit: long division
We may also solve this problem using long division (see the second image). The first step is to look at the leading coefficients: since is equal to
, the first term in the quotient will be
. Since
is equal to
, we must now subtract that from
.
We repeat the same process, as shown in the image. Since we eventually get to zero, the remainder is zero, and the polynomial at the top (x^2-6x+4) is the quotient.
<em>I know this might be difficult to follow, so please comment if you have any questions.</em>
