Synthetic Division

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A series of free, online Intermediate Algebra Lessons or Algebra II lessons.
Videos, worksheets, and activities to help Algebra students.

In this lesson, we will learn

• how to divide polynomials using synthetic division
• how to use synthetic division to evaluate polynomials

Dividing Polynomials using Synthetic Division

When dividing polynomials, we can use either long division or synthetic division to arrive at an answer.
We can use synthetic division to divide polynomials if the degree in the divisor is equal to 1 and if the coefficient of the variable in the divisor is equal to 1.
After we do this, we can write the coefficients of the divisor and dividend and use synthetic division to determine the answer.
Examples:
Divide using synthetic division.
1. (2x³ + 6x² + 29) ÷ (x + 4)
2. (2x³ + 6x² - 17x + 15) ÷ (x + 5)
3. (y⁵ - 32) ÷ (y - 2)
4. (16x³ - 2 + 14x - 12x²) ÷ (2x + 1)

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Synthetic Division - A shortcut for long division
This video shows how you can use synthetic division to divide a polynomial by a linear expression.
It also shows how synthetic division can be used to evaluate polynomials.
Example:
(x³ - 2x² + 3x - 4) ÷ (x - 2)

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Synthetic Division - Example 2
Another video showing how to use synthetic division to divide a polynomial by a linear expression and also how to use the remainder to evaluate the polynomial.
Example:
(x⁴ - x² + 5) ÷ (x + 3)

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Using Synthetic Division to Evaluate Polynomials

We can use synthetic division when dividing or evaluating polynomials. To evaluate polynomials using synthetic division, we use the same process as dividing polynomials with synthetic division. When evaluating a polynomial using synthetic division, the remainder is the answer that you would arrive at if you evaluated by plugging in.

How to use synthetic division to evaluate a function?
Example:
P(x) = x⁴ - 5x² + 4x + 12
Evaluate P(-4)

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How to use Synthetic Division to evaluate a polynomial?
Example:
Use synthetic division to evaluate P(7), where P(x) = 3x⁵ -22x⁴ + 8x³ - 40x + 11. P(7) is the remainder from division of P(x) by (x - 7).

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