# 5.3 – Synthetic Division

Key Terms

• Descending Order – In order from highest power to lowest (exponents), from left to right.
• This is the standard way to write a polynomial.
• Dividend – The number to be divided.
• Divisor – The number being divided by.
• Quotient – The result of a division problem.
• Remainder – A number left over after dividing whole numbers. The letter r is sometimes used to represent a remainder.
• Synthetic Division – A process for dividing polynomials.
• This method is known as the “shortcut” for dividing polynomials because it requires fewer calculations than does long division.
• It ONLY works when the divisor is in linear form: $(x\pm c)$.

Review

• Parts of Division Problems

• Integer Long Division with Fractional Remainder

• Polynomial Long Division (No Remainder)
• $(x^2+3x+2) \div(x+1)$
• x≠-1 because -1+1=0, and you cannot divide by zero (undefined)

Notes

• Synthetic Division
• Always write polynomials in descending order (highest degree exponent first).
• Ex. $3x+2x^4+5-6x^5-x^3+3x^2$ should be rewritten in this order: $-6x^5+2x^4-x^3+3x^2+3x+5$
• If any terms are missing, insert them with a coefficient of zero.
• Ex. $2x^2+5$ should be written $2x^2+0x+5$, where the zero acts as a placeholder for the x-term.
• Synthetic Division is a shortcut for polynomial division
• Below are the STEPS for Solving with Synthetic Division

• Take a look at step 2, above.  The “a” of (x – a) is the divisor .
• Ex. If (x – 2) is your divisor, you would use “2” as your divisor term on the outside.
• Ex. If (x + 4), change it to (x – (-4)), and use “-4” as your divisor term on the outside.
• So, you should notice that you always use the opposite of whatever divisor you are given:
• Ex. (x – 6) would be +6
• Ex. (x + 3) would be -3

• Ex. $(x^2-2x+5) \div(x+4)$
• First, rewrite the divisor as $(x-(-4))$ so that you know your “a” is -4.

• Remainders
• Synthetic division may have fractional remainders, just like integer division

Examples

• Ex 1: Synthetic Division Setup

• Ex 2. Solving with Synthetic Division