factor the trinomial a^2+4ap-32p^2

Question

1.factor the trinomial a^2+4ap-32p^2

 

2. solve the inequality. Graph the solution set and write it in interval notation |x| greater then equal to 12 

 

3.Reduce the rational. Express to lowest terms X^2-49/7-x

 

4. Divide a^3b^3 c+12abc^3+5a^4b^6/6abc^3

Answer 1

1. To factor the trinomial try reverse FOIL followed by guess and check:

What two numbers when multiplied equals -32 but when added equals 4?

(a+8p)(a-4p)

 

2. The inequality is |x|≥12 and this is actually two inequalities: x≥12 and -x≥12 or x≤-12

In interval notation this would be: (-∞,-12]U[12,-∞)

Here is a crude graph:

-∞     -12          12          ∞

<-------•             •--------->

 

3. The rational function can be rewritten as: -(x²-49)/(x-7)

Factoring the numerator gives: -(x+7)(x-7)/(x-7)=-(x+7) or -x-7

 

4. If this is correct: (a³b³c + 12abc³ + 5a⁴b⁶)/(6abc³) then each expression (3 of them) in the numerator must be divided by the denominator to simplify:

a²b²/(6c²) + 2 + 5a³b⁵/(6c³)

Answer 2

1)

 

To factor a trinomial, we use the FOIL method.  FOIL is factoring using two binomial factors.

 

The first term of the trinomial is   a².   The last term of the trinomial is  -32p².  We can set up our binomial factors knowing this fact.  When combine the inner and outer terms, we should get 4ap.

 

(a + p)(a + p)

 

 

Next, we need to find coefficients for the p's so the when combine the inner and outer terms, we should get the term  4ap.

 

(a + 8p)(a - 4p)

 

 

2)

 

| x | ≥ 12

 

The absolute value of any number is always positive.  Therefore, we can break up this inequality into two parts.

 

 

x ≥ 12                 or            x ≤ -12

 

 

Lets check this inequality.  Test x=-13  and x=13 into the inequality.

 

| -13 | ≥ 12                   and              | 13 | ≥ 12

 

13 ≥ 12                         and               13 ≥ 12

 

 

The inequality is true.

 

The solution is all values of x less than or equal -12  and greater than or equal to 12.

 

To graph this solution, draw a number line that contains the number -12 and 12.  Draw closed circles at -12 and 12.  Draw an arrow going in the left direction from -12.  Draw an arrow going in the right direction from 12.

 

 

3)

 

(x² - 49) / (7 - x)

 

 

Notice that (x² - 49) is a difference of perfect squares.  Factoring, we get

 

(x + 7)(x - 7) / (7 - x)

 

 

If we negate the denominator,

 

-(7 - x) =  x - 7

 

 

(x + 7)(x - 7) / [-(7 - x)]

 

 

Cancelling out the (x - 7) terms,

 

-(x + 7) = -x - 7

 

 

Note that we have a negative sign in the solution because we have to manipulate a factor to cancel it out.

 

 

4)

 

(a³b³c + 12abc³ + 5a⁴b⁶c⁰) / (6abc⁴)

 

Note that c⁰ is the same as 1.

 

 

Divide each term in the numerator by the denominator.  Divide the constants and subtract exponents of common bases.

 

(1/3)a²b²c^-3 + 2c^-1 + (5/6)a³b⁵c^-4