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Fields Medal Prize Winners (1998)




TUTORIALS:


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
Fractions
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
Roots
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
Powers
Rewriting Algebraic Fractions
Exponents
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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Factoring Trinomials

Factoring a Trinomial of the Form x2 + bx + c

The product of two binomials can be a trinomial.

For example, let’s multiply (x + 7) by (x + 3).

Use FOIL. (x + 7)(x + 3) = x · x + x · 3 + 7 · x + 7 · 3
Simplify each term.

Combine like terms.

=

=

x2

x2

+

+

3x

 

+

10x

7x

 

+

+

21

21

 

Notice the relationship between the trinomial x2 + 10x + 21 and the binomials (x + 7) and (x + 3):

• The first term of the trinomial, x2, is the product of x and x, the first term of each binomial. (x + 7)(x + 3)
• The coefficient of the middle term of the trinomial is 10, the sum of 3 and 7, the constants in the binomials. = x · x + (7 + 3)x + 7 · 3

= x2 + 10x + 21

• The last term of the trinomial is 21, is the product of 3 and 7, the constants in the binomials.
 

This relationship holds in general. That is, if the product of two binomials (x + r) and (x + s) is a trinomial of the form x2 + bx + c:

• c is the product of r and s.

• b is the sum of r and s.

We use this to factor trinomials of the form x2 + bx + c.

This procedure is called the product-sum method of factoring because we seek two integers whose product is c and whose sum is b.

 

Procedure — To Factor x2 + bx + c (Product-Sum Method)

Step 1 Find two integers whose product is c and whose sum is b.

Step 2 Use the integers from Step 1 as the constants, r and s, in the binomial factors (x + r) and (x + s).

To check the factorization, multiply the binomial factors.

 

Example

Factor: x2 + 3x + 2

Solution

This trinomial has the form x2 + bx + c where b = 3 and c = 2.

Step 1 Find two integers whose product is c and whose sum is b.

Since c is 2, list pairs of integers whose product is 2.

Then, find the sum of each pair of integers.

Product

1 · 2

(-1) · (-2)

Sum

3

-3

 

The first product, 1 · 2, gives the required sum, 3.

Step 2 Use the integers from Step 1 as the constants, r and s, in the binomial factors (x + r) and (x + s).

The result is: x2 + 3x + 2 = (x + 1)(x + 2).

We multiply to check the factorization.

Is (x + 1)(x + 2) = x2 + 3x + 2 ?

Is x2 + 2x + 1x + 2 = x2 + 3x + 2 ?

Is x2 + 3x + 2 = x2 3x + 2 ? Yes

Note:

Multiplication is commutative, so the factorization may also be written: (x + 2)(x + 1).