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)
| 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.
(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.
(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