# How do you use synthetic substitution to evaluate f(-3) for f(x)=2x^3-6x^2-5x+7?

Feb 24, 2017

$f \left(- 3\right) = \textcolor{g r e e n}{- 86}$
$\textcolor{w h i t e}{\text{XXXXXX}}$method using synthetic substitution below:

#### Explanation:

Setup a row (row 1) with the coefficients of the terms of the polynomial arranged in decreasing degree (use $0$'s if any degree is missing).
On a second row, write a $0$ under the first coefficient (indicating the result of any prior evaluation); underline row 2; if you like you can add a plus sign to the left to show that we are going to be adding the numbers in rows 1 and 2.
On a third row, write the argument value that we want to use to evaluate the polynomial; (I like to put a multiplication sign in front to remind me that I'm going to multiply by this number.

{: (,,x^3,x^2,x^1,x^0,), (," | ",2,-6,-5,+7,color(white)("XX")larr"row 1"), (underline(+color(white)("XXX"))," | ",underline(0),underline(color(white)(-6)),underline(color(white)(+36)),underline(color(white)(-93)),color(white)("XX")larr"row 2"), (xx (-3)," | ",color(white)(2),color(white)(-12),color(white)(+31),color(white)(-86),color(white)("XX")larr"row 3") :}

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Add the row 1 number and row 2 number (for the latest column in which we wrote a number in row 2) and write the sum in that column in row 3.
Multiply this number (in row 3) by our argument value and write the product in the next column of row 2.

{: (,,x^3,x^2,x^1,x^0,), (," | ",2,-6,-5,+7,color(white)("XX")larr"row 1"), (underline(+color(white)("XXX"))," | ",underline(0),underline(-6),underline(color(white)(+36)),underline(color(white)(-93)),color(white)("XX")larr"row 2"), (xx (-3)," | ",2,color(white)(-12),color(white)(+31),color(white)(-86),color(white)("XX")larr"row 3") :}

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Repeat this process for each column.
The final column of row 3 contains the value of the polynomial at the given argument.
{: (," | ",2,-6,-5,+7,color(white)("XX")larr"row 1"), (underline(+color(white)("XXX"))," | ",underline(0),underline(-6),underline(+36),underline(-93),color(white)("XX")larr"row 2"), (xx (-3)," | ",2,-12,+31,color(red)(-86),color(white)("XX")larr"row 3") :}