Question 166556: Help me please:
Use synthetic division to find the quotient and remainder (4x^3-6x+5)/(x-3).
Thank you. Found 2 solutions by jim_thompson5910, gonzo:Answer by jim_thompson5910(35256) (Show Source):
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.(note: remember if a polynomial goes from to there is a zero coefficient for . This is simply because really looks like
3
|
4
0
-6
5
|
Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 4)
3
|
4
0
-6
5
|
4
Multiply 3 by 4 and place the product (which is 12) right underneath the second coefficient (which is 0)
3
|
4
0
-6
5
|
12
4
Add 12 and 0 to get 12. Place the sum right underneath 12.
3
|
4
0
-6
5
|
12
4
12
Multiply 3 by 12 and place the product (which is 36) right underneath the third coefficient (which is -6)
3
|
4
0
-6
5
|
12
36
4
12
Add 36 and -6 to get 30. Place the sum right underneath 36.
3
|
4
0
-6
5
|
12
36
4
12
30
Multiply 3 by 30 and place the product (which is 90) right underneath the fourth coefficient (which is 5)
3
|
4
0
-6
5
|
12
36
90
4
12
30
Add 90 and 5 to get 95. Place the sum right underneath 90.
3
|
4
0
-6
5
|
12
36
90
4
12
30
95
Since the last column adds to 95, we have a remainder of 95. This means is not a factor of
Now lets look at the bottom row of coefficients:
The first 3 coefficients (4,12,30) form the quotient
and the last coefficient 95, is the remainder, which is placed over like this
Putting this altogether, we get:
So
which looks like this in remainder form:
remainder 95
You can put this solution on YOUR website! (4x^3-6x+5)/(x-3).
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this is relatively easy to do on paper but it's a real pain in the neck to type it in and have you see how it works.
i'll do my best.
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your original equation is:
4x^3 - 6x + 5
you want to divide that by (x-3)
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make sure all equations are in exponential order form (highest order of exponent on the left, next highest to the right of that, next highest to the right of that, etc.
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look at your highest order exponent in your original equation (the dividend equation).
it is 4x^3
what do you have to multiply (x-3) by to make the highest order of the multiplication equal to 4x^3?
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the answer is 4x^2.
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you multiply (x-3) * 4x^2
you get 4x^3 - 12x^2
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you take your original equation of 4x^3 - 6x + 5 and subtract (4x^3 - 12x^2) from it.
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you get 12x^2 -6x + 5 which is the first remainder.
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you look at the highest order exponent of the first remainder.
it is 12x^2
what do you have to multiply (x-3) by to make the highest order of the multiplication equal to 12x^2?
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the answer is 12x.
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you multiply (x-3) * 12x
you get 12x^2 - 36x
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you take the first remainder of 12x^2 - 6x + 5 and subtract (12x^2 - 36x) from it.
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you get 30x + 5 which is the second remainder.
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you look at the highest order exponent of the second remainder.
it is 30x
what do you have to multiply (x-3) by to make the highest order of the multiplication equal to 30x?
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the answer is 30.
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you multiply (x-3) * 30
you get 30x - 90
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you take the second remainder of 30x + 5 and subtract (30x - 90) from it.
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you get 95 which is the third remainder.
this is also the last remainder because there is nothing left to the right of that.
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your answer is: 4x^2 + 12x + 30 with a remainder of 95.
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the 4x^2 was obtained after the first division into the original equation.
the 12x was obtained after the second division into the first remainder.
the 30 was obtained after the third division into the second remainder.
the 95 is the final remainder.
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to prove your answer is correct, you multiply (4x^2 + 12x + 30) by (x-3)
first multiply (4x^2 + 12x + 30) by -3.
you get
-12x^2 - 36x - 90
then multiply (4x^2 + 12x + 30 by x.
you get
4x^3 + 12x^2 + 30x.
next you add these two terms together.
you get
4x^3 + 12x^2 - 12x^2 + 30x - 36x - 90
combining like terms, you get
4x^3 - 6x - 90
now you have to add the remainder which was 95.
you get
4x^3 - 6x - 90 + 95
combining like terms, you get
4x^3 - 6x + 5
this is the original equation you started from so you know that the division was done properly.
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