How to Do Synthetic Substitution

Synthetic substitution is a shortcut method used in algebra for dividing a polynomial by a binomial of the form (x – c). Here’s a step-by-step guide on how to perform synthetic substitution:

1. Identify the Polynomial and the Value: Start by identifying the polynomial you want to evaluate and the value of ‘c’ from the binomial (x – c).
2. Set Up the Synthetic Division: Write down the coefficients of the polynomial in a row. For example, if your polynomial is 2x^3 – 6x^2 + 3, you would write down 2, -6, and 3. If there are missing degrees, use 0 as the placeholder.
3. Write ‘c’ on the Left: Write the value ‘c’ that you are substituting on the left side of the coefficients. For example, if you are calculating for (x – 2), write 2.
4. Bring Down the Leading Coefficient: Bring down the first coefficient directly below the line.
5. Multiply and Add: Multiply the number you just brought down by ‘c’, and write the result under the next coefficient. Add this result to the coefficient above it. Repeat this step for all coefficients.
6. Read the Result: The final number you get after the last addition is the value of the polynomial when x = c. The other numbers represent the coefficients of the quotient polynomial if you were performing polynomial division.

For example, let’s say we want to evaluate P(x) = 2x^3 – 6x^2 + 3 at x = 2:

• Coefficients: 2, -6, 0, 3 (note the 0 for the missing x-term)
• Set up: 2 goes to the left and you write the coefficients as mentioned.
• Bring down the 2:

     2  |  2  -6  0  3
          |       4
          ---------------
            2  -2
           |      -4
           ---------------
              2   -1  -1

• So, you would multiply and add until you get the answer. The last number gives you the result of P(2).

In summary, synthetic substitution is a powerful tool for quickly evaluating polynomials at specific values, especially when you want to avoid the more tedious long division method.