Thomas's Responses in WyzAnt Answers

A way to check your answer, which is correct from Kevin S., is to multiply both the numerator and denominator by the original denominator (x+2).  As shown:

(4x^2 - 1) / 1 * (x + 2) / (x + 2)  (doing this step is effectively multiplying your answer by "1")

(4x^2 - 1)*(x + 2) / (x+2)

(4x^3 + 8x^2 - x - 2) / (x + 2)  (FOIL)

Since we got the same answer, the solution checks out.  

Polynomials usually higher than 3 degrees is usually solved with synthetic division.  This is usually a trial and error method given the factors that can be multiplied together to form either the beginning and ending coefficients 4 and 35 respectively.  Both negative and positive values of these factors should be tried.

The important process is to have no remainder left over once we attempt synthetic division.  Essentially, the last number we bring down will be zero.  If not, we must try another factor and repeat the process.

Using -5 as a potential factor:

-5 |+4   +21  +33  +147  +35  (notice we used the coefficients in front of each degree)

    _____-20__-5___-140_-35__  (also note there is a place holder for each degree, from 4 down to 0)

     +4   +1   +28   +7       0  (process of synthetic division, bring down the first coefficient (+4), then multiply +4*-5 = -20 and place the -20 on the middle row under the second coefficient (+21).  Then add the top and middle rows +21 + -21 = +1.  Repeat this process.  This is factor since our remainder, the last number on the bottom row, is equal to 0)

So, the first factor or rational zero of this polynomial is -5.  This would be represented as (x + 5)

Now we rewrite the equation and see if is now factorable.  The numbers on the bottom row are the new coefficients of the partially factored polynomial.  Also, the polynomial has decreased by the power of 1 since we factored out an x.

New polynomial: (x+5)(4x³ + 1x² + 28x + 7)

Since this still doesn't look like it's easily factored.  We will have to repeat this process again, with trial and error.  The next factor would -1/4.

Through the process of synthetic division, the polynomial looks like this:

(x+5)(x+1/4)(4x² + 0x + 28)  (notice the polynomial decreased another degree and all degree of x from 2 to 0 have placeholders)

From here we can factor the remaining polynomial:

(x+5)(x+1/4)*4(x² + 7)  (since we are finding values of x that make this polynomial equal to zero, we can set the entire polynomial equal to zero.  From here, we can divide both sides by 4)

(x+5)(x+1/4)(x²+7) = 0  (finish factoring the x²)

(x+5)(x+1/4)(x+√(-7))(x-√(-7)) = 0

Since there are 4 degrees in the original polynomial, there must be 4 answers or roots for x that make this equation equal to zero.

x = -5, -1/4, -i√7, +i√7 (since the last two roots are not real numbers)

answer:  x = -5, -1/4

first multiply out the constants in front of each parentheses:

30z - 6 - 4z - 12 = 3z + 3 (simplify)

26z - 18 = 3z + 3 (subtract 3z from both sides)

23z - 18 = 3 (add 18 to each side)

23z = 21 (divide both side by 23)

z = 21/23

remember with equations, what ever we do to one side of the equation, we have to do to the other.  this keeps everything equal.

To check this answer, we plug 21/23 back into the original equation:

30*(21/23) - 6 - 4*(21/23) - 12 = 3*(21/23) + 3

Order of operations is important here and being comfortable with fractions:

630/23 - 6 - 84/23 - 12 = 63/23 + 3 (now find the LCD so we can add these fractions and whole integers) (multiply the non fractions by 23 ) (LCD = 23)

630/23 - 138/23 - 84/23 - 276/23 = 63/23 + 69/23 (now add or subtract, just remember to keep the bottom (denominator) the same at 23)

132/23 = 132/23  the solution checks out

Constants are simply a number.  A value that is considered constant.  example 1, 2, or 3.

As an example:

Variables are symbols for values that can change, such as y = x^2, where we have two variables x and y.  x isn't defined as a specific number or value, so it can change.  y is directly associated with x's value, so it changes as well.

if we rewrite the equation to say y = 2x^2 + 9, 2 and 9 are constants.  we still have the two variables, x and y, but 2 and 9 will always be 2 and 9.  they don't change or can't be substituted for different values.

Start with hypothesis test:

Ho: pi = 0.18  (pi = population proportion) (null hypothesis, what is expected from past observations)

Ha: pi not= 0.18  (alternative hypothesis, what we are trying to prove, something has changed)

Then use the confidence interval for proportion:

p +/- Za/2 * sqrt((p*(1-p))/n)

p = x / n = 27 / 100 = 0.27  (sample proportion)

Za/2 = 1.96  (Z statistic used for a two tailed test at 95% level of confidence)

n = 100  (sample size)

Plug in values into formula:

0.27 +/- 1.96*sqrt((0.27*(1-0.27))/100)  (simplify)

0.27 +/- 1.96*0.0444

0.27 +/- 0.087 = 0.357 and 0.183  (these are the upper and lower values for the confidence interval)

Answer: We have a 95% level of confidence that the true population proportion of adults tested for HIV is between 18.3% and 35.7%. Since the previously observed proportion of 18% falls outside of the lower limit of this confidence interval (18.3%), we reject the null hypothesis (Ho) and accept the alternative (Ha). The proportion of adults tested for HIV has changed (not equal to 18%).