SOLUTION: Thank you for looking... Multiply (2x - 3y)^2 Multiply (6x - 7)(6x + 7) Divide (3x^3 - x^2 + 10x - 4) / (x + 3)

Algebra ->  Exponents -> SOLUTION: Thank you for looking... Multiply (2x - 3y)^2 Multiply (6x - 7)(6x + 7) Divide (3x^3 - x^2 + 10x - 4) / (x + 3)      Log On


   

Question 196043: Thank you for looking...
Multiply (2x - 3y)^2
Multiply (6x - 7)(6x + 7)
Divide (3x^3 - x^2 + 10x - 4) / (x + 3)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
# 1



%282x-3y%29%5E2 Start with the given expression.


%282x-3y%29%282x-3y%29 Expand. Remember something like x%5E2=x%2Ax.


Now let's FOIL the expression.


Remember, when you FOIL an expression, you follow this procedure:


%28highlight%282x%29-3y%29%28highlight%282x%29-3y%29 Multiply the First terms:%282%2Ax%29%2A%282%2Ax%29=4%2Ax%5E2.


%28highlight%282x%29-3y%29%282x%2Bhighlight%28-3y%29%29 Multiply the Outer terms:%282%2Ax%29%2A%28-3%2Ay%29=-6%2Ax%2Ay.


%282x%2Bhighlight%28-3y%29%29%28highlight%282x%29-3y%29 Multiply the Inner terms:%28-3%2Ay%29%2A%282%2Ax%29=-6%2Ax%2Ay.


%282x%2Bhighlight%28-3y%29%29%282x%2Bhighlight%28-3y%29%29 Multiply the Last terms:%28-3%2Ay%29%2A%28-3%2Ay%29=9%2Ay%5E2.


4%2Ax%5E2-6%2Ax%2Ay-6%2Ax%2Ay%2B9%2Ay%5E2 Now collect every term to make a single expression.


4%2Ax%5E2-12%2Ax%2Ay%2B9%2Ay%5E2 Now combine like terms.


So %282x-3y%29%5E2 FOILs to 4%2Ax%5E2-12%2Ax%2Ay%2B9%2Ay%5E2.


In other words, %282x-3y%29%5E2=4%2Ax%5E2-12%2Ax%2Ay%2B9%2Ay%5E2.




# 2



%286x-7%29%286x%2B7%29 Start with the given expression.


Now let's FOIL the expression.


Remember, when you FOIL an expression, you follow this procedure:


%28highlight%286x%29-7%29%28highlight%286x%29%2B7%29 Multiply the First terms:%286%2Ax%29%2A%286%2Ax%29=36%2Ax%5E2.


%28highlight%286x%29-7%29%286x%2Bhighlight%287%29%29 Multiply the Outer terms:%286%2Ax%29%2A%287%29=42%2Ax.


%286x%2Bhighlight%28-7%29%29%28highlight%286x%29%2B7%29 Multiply the Inner terms:%28-7%29%2A%286%2Ax%29=-42%2Ax.


%286x%2Bhighlight%28-7%29%29%286x%2Bhighlight%287%29%29 Multiply the Last terms:%28-7%29%2A%287%29=-49.


36%2Ax%5E2%2B42%2Ax-42%2Ax-49 Now collect every term to make a single expression.


36%2Ax%5E2-49 Now combine like terms.


So %286x-7%29%286x%2B7%29 FOILs to 36%2Ax%5E2-49.


In other words, %286x-7%29%286x%2B7%29=36%2Ax%5E2-49.


Note: this is a difference of squares.



# 3

Let's simplify this expression using synthetic division


Start with the given expression %283x%5E3+-+x%5E2+%2B+10x+-+4%29%2F%28x%2B3%29

First lets find our test zero:

x%2B3=0 Set the denominator x%2B3 equal to zero

x=-3 Solve for x.

so our test zero is -3


Now set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of the numerator to the right of the test zero.
-3|3-110-4
|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 3)
-3|3-110-4
|
3

Multiply -3 by 3 and place the product (which is -9) right underneath the second coefficient (which is -1)
-3|3-110-4
|-9
3

Add -9 and -1 to get -10. Place the sum right underneath -9.
-3|3-110-4
|-9
3-10

Multiply -3 by -10 and place the product (which is 30) right underneath the third coefficient (which is 10)
-3|3-110-4
|-930
3-10

Add 30 and 10 to get 40. Place the sum right underneath 30.
-3|3-110-4
|-930
3-1040

Multiply -3 by 40 and place the product (which is -120) right underneath the fourth coefficient (which is -4)
-3|3-110-4
|-930-120
3-1040

Add -120 and -4 to get -124. Place the sum right underneath -120.
-3|3-110-4
|-930-120
3-1040-124

Since the last column adds to -124, we have a remainder of -124. This means x%2B3 is not a factor of 3x%5E3+-+x%5E2+%2B+10x+-+4
Now lets look at the bottom row of coefficients:

The first 3 coefficients (3,-10,40) form the quotient

3x%5E2+-+10x+%2B+40

and the last coefficient -124, is the remainder, which is placed over x%2B3 like this

-124%2F%28x%2B3%29



Putting this altogether, we get:

3x%5E2+-+10x+%2B+40-124%2F%28x%2B3%29

So %283x%5E3+-+x%5E2+%2B+10x+-+4%29%2F%28x%2B3%29=3x%5E2+-+10x+%2B+40-124%2F%28x%2B3%29

which looks like this in remainder form:

%283x%5E3+-+x%5E2+%2B+10x+-+4%29%2F%28x%2B3%29=3x%5E2+-+10x+%2B+40 remainder -124


You can use this online polynomial division calculator to check your work