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Polynomial Division: Long Division vs. Synthetic Division

Learn the simplified process of synthetic division for dividing polynomials. Understand when to use long division and synthetic division with examples and reminders. Discover the remainder theorem and practice solving problems using both methods.

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Polynomial Division: Long Division vs. Synthetic Division

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  1. Chapter 6 Section 3 Dividing Polynomials

  2. Long Division Vocabulary Reminders

  3. Remember Long Division • Does 8 go into 6? • No • Does 8 go into 64? • Yes, write the integer on top. • Multiply 8∙8 • Write under the dividend • Subtract and Carry Down • How many times does 8 go into 7 evenly? • 0 write over the 7 • Multiply 0∙8 • Subtract and write remainder as a fraction.

  4. The divisor and quotient are only FACTORS if the remainder is Zero.

  5. Examples with variables

  6. Examples • If the divisor has more than one term, always use the term with the highest degree. • A remainder occurs when the degree of the dividend is less than the degree of the divisor

  7. Example:

  8. Try These ExamplesDivide using long division.

  9. Long division of polynomials is tedious! • Lets learn a simplified process!This process is called Synthetic Division • p. 316It may look complicated, but watch a few examples and you will get the hang of it.

  10. Use synthetic division to divide 3x3-4x2+2x-1 by x+1 • Reverse the sign of the constant term in the divisor.Write the coefficients of the polynomial in standard form (Remember to include zeros) • Translation: Instead of write • Bring down the first coefficient • Multiply the first coefficient by the new divisor. Add. • Repeat step 3 until the end. The last number is the remainder. • NOW write the polynomial. • To write the answer use one less degree than the original polynomial. -1 3 -4 2 -1

  11. Example: Use synthetic division to divide • x3+4x2+x-6 by x+1 • x3-2x2-5x+6 by x+2

  12. Remainder Theorem • If a polynomial is being divided by (x-a) then the remainder is P(a). • Example: Use the remainder theorem to find P(-4) for P(x)=x3-5x2+4x+12 • DO NOT change the number P(a) to -a

  13. Try This Problem • Use synthetic division to find P(-1) for P(x)=4x4+6x3-5x2-60

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