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Dividing Polynomials: Long Division and Synthetic Division - Prof. Raya Ojeda, Study notes of Mathematics

A comprehensive guide to dividing polynomials using both long division and synthetic division methods. It explains the concepts, steps involved, and provides illustrative examples to solidify understanding. The document also includes practice problems to reinforce learning and enhance problem-solving skills.

Typology: Study notes

2023/2024

Uploaded on 12/11/2024

miabella-correa
miabella-correa 🇺🇸

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3.5 DIVIDING POLYNOMIALS
Using Long Division to Divide Polynomials
Division of polynomials that contain more than one term has similarities
to long division of whole numbers. We can write a polynomial dividend
as the product of the divisor and the quotient added to the remainder.
The terms of the polynomial division correspond to the digits (and place
values) of the whole number division. This method allows us to divide
two polynomials.
EX. Divide 2x3 − 3x2 + 4x + 5 by x + 2 using the long division
algorithm.
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3.5 DIVIDING POLYNOMIALS

Using Long Division to Divide Polynomials Division of polynomials that contain more than one term has similarities to long division of whole numbers. We can write a polynomial dividend as the product of the divisor and the quotient added to the remainder. The terms of the polynomial division correspond to the digits (and place values) of the whole number division. This method allows us to divide two polynomials. EX. Divide 2x^3 − 3x^2 + 4x + 5 by x + 2 using the long division algorithm.

Using Synthetic Division to Divide Polynomials Synthetic division is a shortcut that can be used when the divisor is a binomial in the form x − k. In synthetic division, only the coefficients are used in the division process. EX. Divide 2x 3 − 3x 2

  • 4x + 5 by x + 2 using the long division algorithm and then by Synthetic Division.

Try It # Divide 16x^3 − 12x^2 + 20x − 3 by 4x + 5.

EX. Use synthetic division to divide −9x 4

  • 10x 3
  • 7x 2 − 6 by x − 1.

EX. The volume of a rectangular solid is given by the polynomial 3x^4 − 3x^3 − 33x^2 + 54x. The length of the solid is given by 3x and the width is given by x − 2. Find the height of the solid.

Try It # The area of a rectangle is given by 3x^3 + 14x^2 − 23x + 6. The width of the rectangle is given by x + 6. Find an expression for the length of the rectangle.