How to Divide Polynomials Using Synthetic Division: 12 Steps

2024 Author: Jason Gerald | [email protected]. Last modified: 2024-01-16 19:04

Synthetic division is a shorthand way of dividing polynomials where you can divide the coefficients of the polynomial by removing the variables and their exponents. This method allows you to keep adding throughout the process, without any subtraction, as you would normally do with traditional division. If you want to know how to divide polynomials using synthetic division, just follow these steps.

Step

Step 1. Write down the problem

For this example, you will divide x³ + 2x² - 4x + 8 where x + 2. Write the equation of the first polynomial, the equation to be divided, in the numerator and write the second equation, the equation that divides, in the denominator.

Step 2. Invert the sign of the constant in the divisor equation

The constant in the divisor equation, x + 2, is positive 2, so the reciprocal of its sign is -2.

Step 3. Write this number outside the inverse division symbol

The inverted division symbol looks like an inverted L. Put the number -2 to the left of this symbol.

Step 4. Write down all the coefficients of the equation to be divided in the division symbol

Write the numbers from left to right like the equation. The result is like this: -2| 1 2 -4 8.

Step 5. Derive the first coefficient

Lower the first coefficient, 1, below it. The result will look like this:

• -2| 1 2 -4 8

  ↓

  1

Step 6. Multiply the first coefficient by the divisor and place it under the second coefficient

Just multiply 1 by -2 to make -2 and write the product under the second part, 2. The result will look like this:

• -2| 1 2 -4 8

  -2

  1

Step 7. Add up the second coefficient with the product and write the answer under the product

Now, take the second coefficient, 2, and add it to -2. The result is 0. Write the result under the two numbers, as you would with long division. The result will look like this:

• -2| 1 2 -4 8

  -2

  1 0

Step 8. Multiply the sum by the divisor and place the result under the second coefficient

Now, take the sum, 0, and multiply it by the divisor, -2. The result is 0. Put this number under 4, the third coefficient. The result will look like this:

• -2| 1 2 -4 8

  -2 0

  1

Step 9. Add up the product and the coefficients of the three and write the result under the product

Add 0 and -4 to -4 and write the answer under 0. The result will look like this:

• -2| 1 2 -4 8

  -2 0

  1 0 -4

Step 10. Multiply this number by the divisor, write it under the last coefficient, and add it by the coefficient

Now, multiply -4 by -2 to make 8, write the answer under the fourth coefficient, 8, and add up the answer by the fourth coefficient. 8 + 8 = 16, so this is your remainder. Write this number under the multiplication result. The result will look like this:

• -2| 1 2 -4 8

  -2 0 8

  1 0 -4 |16

Step 11. Place each new coefficient next to the variable that has a power one level lower than the original variable

In this problem, the result of the first addition, 1, is placed next to x to the power of 2 (one level lower than the power of 3). The second sum, 0, is placed next to x, but the result is zero, so you can omit this part. And the third coefficient, -4, becomes a constant, a number with no variables, because the initial variable is x. You can write an R next to 16 because this number is the remainder of the division. The result will look like this:

• -2| 1 2 -4 8

  -2 0 8

  1 0 -4 |16

  x ² + 0 x - 4 R 16

  x ² - 4 R16

Step 12. Write down the final answer

The final answer is the new polynomial, x² - 4, plus the remainder, 16, divided by the original divisor equation, x + 2. The result will look like this: x² - 4 +16/(x +2).

Tips

• To check your answer, multiply the quotient by the divisor equation and add the remainder. It should be the same as your original polynomial.

  (divisor)(quote)+(remainder)

  (x + 2)(x ² - 4) + 16

  Multiply.

  (x ³ - 4x + 2x ² - 8) + 16

  x ³ + 2 x ² - 4 x - 8 + 16

  x ³ + 2 x ² - 4 x + 8