Question 194080
There are two ways to do this:



Direct Substitution and Evaluation Method:


{{{P(x)=3x^5-8x^4+3x^3+2x^2-16x+14}}} Start with the given equation.



{{{P(3)=3(3)^5-8(3)^4+3(3)^3+2(3)^2-16(3)+14}}} Plug in {{{x=3}}}.



{{{P(3)=3(243)-8(3)^4+3(3)^3+2(3)^2-16(3)+14}}} Raise  {{{3}}} to the 5th power to get {{{243}}}.



{{{P(3)=3(243)-8(81)+3(3)^3+2(3)^2-16(3)+14}}} Raise  {{{3}}} to the 4th power to get {{{81}}}.



{{{P(3)=3(243)-8(81)+3(27)+2(3)^2-16(3)+14}}} Cube {{{3}}} to get {{{27}}}.



{{{P(3)=3(243)-8(81)+3(27)+2(9)-16(3)+14}}} Square {{{3}}} to get {{{9}}}.



{{{P(3)=729-8(81)+3(27)+2(9)-16(3)+14}}} Multiply {{{3}}} and {{{243}}} to get {{{729}}}.



{{{P(3)=729-648+3(27)+2(9)-16(3)+14}}} Multiply {{{-8}}} and {{{81}}} to get {{{-648}}}.



{{{P(3)=729-648+81+2(9)-16(3)+14}}} Multiply {{{3}}} and {{{27}}} to get {{{81}}}.



{{{P(3)=729-648+81+18-16(3)+14}}} Multiply {{{2}}} and {{{9}}} to get {{{18}}}.



{{{P(3)=729-648+81+18-48+14}}} Multiply {{{-16}}} and {{{3}}} to get {{{-48}}}.



{{{P(3)=146}}} Combine like terms.



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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.<TABLE cellpadding=10><TR><TD>3</TD><TD>|</TD><TD>3</TD><TD>-8</TD><TD>3</TD><TD>2</TD><TD>-16</TD><TD>14</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

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

<TABLE cellpadding=10><TR><TD>3</TD><TD>|</TD><TD>3</TD><TD>-8</TD><TD>3</TD><TD>2</TD><TD>-16</TD><TD>14</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>3</TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

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

    <TABLE cellpadding=10><TR><TD>3</TD><TD>|</TD><TD>3</TD><TD>-8</TD><TD>3</TD><TD>2</TD><TD>-16</TD><TD>14</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>9</TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>3</TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Add 9 and -8 to get 1. Place the sum right underneath 9.

    <TABLE cellpadding=10><TR><TD>3</TD><TD>|</TD><TD>3</TD><TD>-8</TD><TD>3</TD><TD>2</TD><TD>-16</TD><TD>14</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>9</TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>3</TD><TD>1</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Multiply 3 by 1 and place the product (which is 3)  right underneath the third  coefficient (which is 3)

    <TABLE cellpadding=10><TR><TD>3</TD><TD>|</TD><TD>3</TD><TD>-8</TD><TD>3</TD><TD>2</TD><TD>-16</TD><TD>14</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>9</TD><TD>3</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>3</TD><TD>1</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Add 3 and 3 to get 6. Place the sum right underneath 3.

    <TABLE cellpadding=10><TR><TD>3</TD><TD>|</TD><TD>3</TD><TD>-8</TD><TD>3</TD><TD>2</TD><TD>-16</TD><TD>14</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>9</TD><TD>3</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>3</TD><TD>1</TD><TD>6</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Multiply 3 by 6 and place the product (which is 18)  right underneath the fourth  coefficient (which is 2)

    <TABLE cellpadding=10><TR><TD>3</TD><TD>|</TD><TD>3</TD><TD>-8</TD><TD>3</TD><TD>2</TD><TD>-16</TD><TD>14</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>9</TD><TD>3</TD><TD>18</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>3</TD><TD>1</TD><TD>6</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Add 18 and 2 to get 20. Place the sum right underneath 18.

    <TABLE cellpadding=10><TR><TD>3</TD><TD>|</TD><TD>3</TD><TD>-8</TD><TD>3</TD><TD>2</TD><TD>-16</TD><TD>14</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>9</TD><TD>3</TD><TD>18</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>3</TD><TD>1</TD><TD>6</TD><TD>20</TD><TD></TD><TD></TD></TR></TABLE>

    Multiply 3 by 20 and place the product (which is 60)  right underneath the fifth  coefficient (which is -16)

    <TABLE cellpadding=10><TR><TD>3</TD><TD>|</TD><TD>3</TD><TD>-8</TD><TD>3</TD><TD>2</TD><TD>-16</TD><TD>14</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>9</TD><TD>3</TD><TD>18</TD><TD>60</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>3</TD><TD>1</TD><TD>6</TD><TD>20</TD><TD></TD><TD></TD></TR></TABLE>

    Add 60 and -16 to get 44. Place the sum right underneath 60.

    <TABLE cellpadding=10><TR><TD>3</TD><TD>|</TD><TD>3</TD><TD>-8</TD><TD>3</TD><TD>2</TD><TD>-16</TD><TD>14</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>9</TD><TD>3</TD><TD>18</TD><TD>60</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>3</TD><TD>1</TD><TD>6</TD><TD>20</TD><TD>44</TD><TD></TD></TR></TABLE>

    Multiply 3 by 44 and place the product (which is 132)  right underneath the sixth  coefficient (which is 14)

    <TABLE cellpadding=10><TR><TD>3</TD><TD>|</TD><TD>3</TD><TD>-8</TD><TD>3</TD><TD>2</TD><TD>-16</TD><TD>14</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>9</TD><TD>3</TD><TD>18</TD><TD>60</TD><TD>132</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>3</TD><TD>1</TD><TD>6</TD><TD>20</TD><TD>44</TD><TD></TD></TR></TABLE>

    Add 132 and 14 to get 146. Place the sum right underneath 132.

    <TABLE cellpadding=10><TR><TD>3</TD><TD>|</TD><TD>3</TD><TD>-8</TD><TD>3</TD><TD>2</TD><TD>-16</TD><TD>14</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>9</TD><TD>3</TD><TD>18</TD><TD>60</TD><TD>132</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>3</TD><TD>1</TD><TD>6</TD><TD>20</TD><TD>44</TD><TD>146</TD></TR></TABLE>


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



So according to the remainder theorem, this means that {{{P(3)=146}}}