Fields Medal Prize Winners (1998)


Adding and Subtracting Monomials
Solving Quadratic Equations by Using the Quadratic Formula
Addition with Negative Numbers
Solving Linear Systems of Equations by Elimination
Rational Exponents
Solving Quadratic Inequalities
Systems of Equations That Have No Solution or Infinitely Many Solutions
Dividing Polynomials by Monomials and Binomials
Polar Representation of Complex Numbers
Solving Equations with Fractions
Quadratic Expressions Completing Squares
Graphing Linear Inequalities
Square Roots of Negative Complex Numbers
Simplifying Square Roots
The Equation of a Circle
Fractional Exponents
Finding the Least Common Denominator
Simplifying Square Roots That Contain Whole Numbers
Solving Quadratic Equations by Completing the Square
Graphing Exponential Functions
Decimals and Fractions
Adding and Subtracting Fractions
Adding and Subtracting Rational Expressions with Unlike Denominators
Quadratic Equations with Imaginary Solutions
Graphing Solutions of Inequalities
FOIL Multiplying Polynomials
Multiplying and Dividing Monomials
Order and Inequalities
Exponents and Polynomials
Variables and Expressions
Multiplying by 14443
Dividing Rational Expressions
Division Property of Radicals
Equations of a Line - Point-Slope Form
Rationalizing the Denominator
Imaginary Solutions to Equations
Multiplying Polynomials
Multiplying Monomials
Adding Fractions
Rationalizing the Denominator
Rational Expressions
Ratios and Proportions
Rationalizing the Denominator
Like Radical Terms
Adding and Subtracting Rational Expressions With Different Denominators
Percents and Fractions
Reducing Fractions to Lowest Terms
Subtracting Mixed Numbers with Renaming
Simplifying Square Roots That Contain Variables
Factors and Prime Numbers
Rules for Integral Exponents
Multiplying Monomials
Graphing an Inverse Function
Factoring Quadratic Expressions
Solving Quadratic Inequalities
Factoring Polynomials
Multiplying Radicals
Simplifying Fractions 1
Graphing Compound Inequalities
Rationalizing the Denominator
Simplifying Products and Quotients Involving Square Roots
Standard Form of a Line
Multiplication by 572
Adding and Subtracting Fractions
Multiplying Polynomials
Factoring Trinomials
Solving Exponential Equations
Solving Equations with Fractions
Simplifying Complex Fractions
Multiplying and Dividing Fractions
Mathematical Terms
Solving Quadratic Equations by Factoring
Factoring General Polynomials
Adding Rational Expressions with the Same Denominator
The Trigonometric Functions
Solving Nonlinear Equations by Factoring
Solving Systems of Equations
Midpoint of a Line Segment
Complex Numbers
Graphing Systems of Equations
Reducing Rational Expressions
Rewriting Algebraic Fractions
Rationalizing the Denominator
Adding, Subtracting and Multiplying Polynomials
Radical Notation
Solving Radical Equations
Positive Integral Divisors
Solving Rational Equations
Rational Exponents
Mathematical Terms
Rationalizing the Denominator
Subtracting Rational Expressions with the Same Denominator
Axis of Symmetry and Vertex of a Parabola
Simple Partial Fractions
Simplifying Radicals
Powers of Complex Numbers
Fields Medal Prize Winners (1998)

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Dividing Polynomials by Monomials and Binomials

Dividing a Polynomial by a Monomial

We know how to divide some simple polynomials by monomials. For example,

We use the distributive property to take one-half of 6x and one-half of 8 to get 3x + 4. So both 6x and 8 are divided by 2. To divide any polynomial by a monomial, we divide each term of the polynomial by the monomial.


Example 1

Dividing a polynomial by a monomial

Find the quotient for (-8x6 + 12x4 - 4x2) ÷ (4x2).


The quotient is -2x4 + 3x2 - 1. We can check by multiplying.

4x2(-2x4 + 3x2 - 1) = -8x6 + 12x4 - 4x2.

Because division by zero is undefined, we will always assume that the divisor is nonzero in any quotient involving variables. For example, the division in Example 1 is valid only if 4x2 ≠ 0, or x2 ≠ 0.


Dividing a Polynomial by a Binomial

Division of whole numbers is often done with a procedure called long division. For example, 253 is divided by 7 as folows:

Note that 36 · 7 + 1 = 253. It is always true that

(quotient)(divisor) + (remainder) = (dividend).

To divide a polynomial by a binomial, we perform the division like long division of whole numbers. For example, to divide x2 - 3x - 10 by x + 2, we get the first term of the quotient by dividing the first term of x + 2 into the first term of x2 - 3x - 10. So divide x2 by x to get x, then multiply and subtract as follows:

1) Divide:

2) Multiply:

3) Subtract:

x2 ÷ x = x

x · (x + 2) = x2 + 2x

-3x - 2x = -5x

Now bring down -10 and continue the process. We get the second term of the quotient (below) by dividing the first term of x + 2 into the first term of -5x - 10. So divide -5x by x to get -5:

1) Divide:

2) Multiply:


3) Subtract:

-5x ÷ x = -5

Bring down -10.

-5(x + 2) = -5x - 10

-10 - (-10) = 0

So the quotient is x - 5, and the remainder is 0.

In the next example we must rearrange the dividend before dividing.


Example 2

Dividing a polynomial by a binomial

Divide 2x3 - 4 - 7x2 by 2x - 3, and identify the quotient and the remainder.


Rearrange the dividend as 2x3 - 7x2  - 4. because the x-term in the dividend is missing, we write 0 · x for it:

The quotient is x2 - 2x - 3, and the remainder is -13. Note that the degree of the remainder is 0 and the degree of the divisor is 1. To check, we must verify that

(2x - 3)(x2 - 2x - 3)  - 13 = 2x3 - 7x2  - 4.


To avoid errors, always write  the terms of the divisor and the dividend in descending order of the exponents and insert a zero for any term that is missing.

If we divide both sides of the equation

dividend = (quotient)(divisor) + (remainder)

by te divisor, we get the equation

This fact is used in expressing improper fractions as mixed numbers. For example, if 19 is divided by 5, the quotient is 3 and the remainder is 4. So

We can also use this form to rewrite algebraic fractions.