SOLUTION: If P(x) = 3x^5 - 8x^4 + 3x^3 + 2x^2 - 16x + 14, then P(3) = ? ...is it 682?

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Question 194080This question is from textbook Saxon Algebra 2
: If P(x) = 3x^5 - 8x^4 + 3x^3 + 2x^2 - 16x + 14, then P(3) = ?
...is it 682?
This question is from textbook Saxon Algebra 2

Found 2 solutions by Edwin McCravy, jim_thompson5910:
Answer by Edwin McCravy(8908) About Me  (Show Source):
You can put this solution on YOUR website!
If P(x) = 3x^5 - 8x^4 + 3x^3 + 2x^2 - 16x + 14, then P(3) = ?
...is it 682?

No it isn't.

There are two ways to find P(3).

Method 1.  Substitute 3 for x in

P%28x%29+=+3x%5E5+-+8x%5E4+%2B+3x%5E3+%2B+2x%5E2+-+16x+%2B+14

P%283%29+=+3%283%29%5E5+-+8%283%29%5E4+%2B+3%283%29%5E3+%2B+2%283%29%5E2+-+16%283%29+%2B+14

P%283%29+=+3%28243%29+-+8%2881%29+%2B+3%2827%29+%2B+2%289%29+-+48+%2B+14

P%283%29+=+729+-+648+%2B+81+%2B+18+-+48+%2B+14

P%283%29+=+146

Method 2 (Much easier, by synthetic division).

Start with this:

3 | 3 -8  3   2 -16   14
  |     
   ---------------------
   
and end up with this:

3 | 3 -8  3   2 -16   14
  |    9  3  18  60  132 
   ---------------------
    3  1  6  20  44  146

The answer, 146, is in the lower right 
corner of the synthetic division.

Edwin

Answer by jim_thompson5910(28593) About Me  (Show Source):
You can put this solution on YOUR website!
There are two ways to do this:


Direct Substitution and Evaluation Method:

P%28x%29=3x%5E5-8x%5E4%2B3x%5E3%2B2x%5E2-16x%2B14 Start with the given equation.


P%283%29=3%283%29%5E5-8%283%29%5E4%2B3%283%29%5E3%2B2%283%29%5E2-16%283%29%2B14 Plug in x=3.


P%283%29=3%28243%29-8%283%29%5E4%2B3%283%29%5E3%2B2%283%29%5E2-16%283%29%2B14 Raise 3 to the 5th power to get 243.


P%283%29=3%28243%29-8%2881%29%2B3%283%29%5E3%2B2%283%29%5E2-16%283%29%2B14 Raise 3 to the 4th power to get 81.


P%283%29=3%28243%29-8%2881%29%2B3%2827%29%2B2%283%29%5E2-16%283%29%2B14 Cube 3 to get 27.


P%283%29=3%28243%29-8%2881%29%2B3%2827%29%2B2%289%29-16%283%29%2B14 Square 3 to get 9.


P%283%29=729-8%2881%29%2B3%2827%29%2B2%289%29-16%283%29%2B14 Multiply 3 and 243 to get 729.


P%283%29=729-648%2B3%2827%29%2B2%289%29-16%283%29%2B14 Multiply -8 and 81 to get -648.


P%283%29=729-648%2B81%2B2%289%29-16%283%29%2B14 Multiply 3 and 27 to get 81.


P%283%29=729-648%2B81%2B18-16%283%29%2B14 Multiply 2 and 9 to get 18.


P%283%29=729-648%2B81%2B18-48%2B14 Multiply -16 and 3 to get -48.


P%283%29=146 Combine like terms.


--------------------------------------------------------------------------------

OR....

Synthetic Division Method:


First lets find our test zero:

x-3=0 Set the denominator x-3 equal to zero

x=3 Solve for x.

so our test zero is 3


Now set up a synthetic division table by placing the test zero in the upper left corner and placing the coefficients of the function to the right of the test zero.
3|3-832-1614
|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 3)
3|3-832-1614
|
3

Multiply 3 by 3 and place the product (which is 9) right underneath the second coefficient (which is -8)
3|3-832-1614
|9
3

Add 9 and -8 to get 1. Place the sum right underneath 9.
3|3-832-1614
|9
31

Multiply 3 by 1 and place the product (which is 3) right underneath the third coefficient (which is 3)
3|3-832-1614
|93
31

Add 3 and 3 to get 6. Place the sum right underneath 3.
3|3-832-1614
|93
316

Multiply 3 by 6 and place the product (which is 18) right underneath the fourth coefficient (which is 2)
3|3-832-1614
|9318
316

Add 18 and 2 to get 20. Place the sum right underneath 18.
3|3-832-1614
|9318
31620

Multiply 3 by 20 and place the product (which is 60) right underneath the fifth coefficient (which is -16)
3|3-832-1614
|931860
31620

Add 60 and -16 to get 44. Place the sum right underneath 60.
3|3-832-1614
|931860
3162044

Multiply 3 by 44 and place the product (which is 132) right underneath the sixth coefficient (which is 14)
3|3-832-1614
|931860132
3162044

Add 132 and 14 to get 146. Place the sum right underneath 132.
3|3-832-1614
|931860132
3162044146


Since the last column adds to 146, we have a remainder of 146.


So according to the remainder theorem, this means that P%283%29=146