Vocabulary
Monomial- an expression that is either a number, a variable, or the product of a number and one or more variables.
Binomial- the sum of two monomials.
Trinomial- the sum of three monomials.
Quadratic Equation- an equation that can be written in the form ax^2+bx+c=0 where a does not equal 0.
Root of an Equation- the solutions of a quadratic equation are its roots.
Zero of a Function- a number k is a zero of a function f if f(k)=0.
Binomial- the sum of two monomials.
Trinomial- the sum of three monomials.
Quadratic Equation- an equation that can be written in the form ax^2+bx+c=0 where a does not equal 0.
Root of an Equation- the solutions of a quadratic equation are its roots.
Zero of a Function- a number k is a zero of a function f if f(k)=0.
Essential Information
FOIL can be used to write (x+4)(x+7) as x^2+11x+28. It can also be used to write the trinomial as a product of 2 binomials.
x^2+bx+c=(x+m)(x+n)
=x^2=(m+n)+mn
m+n=b mn=c
x^2+bx+c=(x+m)(x+n)
=x^2=(m+n)+mn
m+n=b mn=c
Factoring Special Products
Factoring quadratic expressions often involves trial and error. Some expressions are easy to factor because of the special patterns they follow.
Special Factoring Patterns:
Difference of Two Squares a^2-b^2=(a-b)(a+b) x^2-4=(x+2)(x-2)
Perfect Square Trinomial a^2+2ab+b^2=(a+b)^2 x^2+6x+9=(x+3)^2
a^2-2ab+b^2=(a-b)^2 x^2-4x=4=(x-2)^2
Solving Quadratic Equations
Factoring can be used to solve certain quadratic equations. The standard form is ax^2+bx+c=0, where a is not equal to zero. The solutions are called roots, and can be found using the zero product property.
Zero Product Property:
Zeros of a Function
The x-intercepts of the graph of y=a(x-p)(x-q) are p and q. Because the function's value is zero when x = p and when x = q, the numbers p and q are called zeros of the functions
Factoring quadratic expressions often involves trial and error. Some expressions are easy to factor because of the special patterns they follow.
Special Factoring Patterns:
Difference of Two Squares a^2-b^2=(a-b)(a+b) x^2-4=(x+2)(x-2)
Perfect Square Trinomial a^2+2ab+b^2=(a+b)^2 x^2+6x+9=(x+3)^2
a^2-2ab+b^2=(a-b)^2 x^2-4x=4=(x-2)^2
Solving Quadratic Equations
Factoring can be used to solve certain quadratic equations. The standard form is ax^2+bx+c=0, where a is not equal to zero. The solutions are called roots, and can be found using the zero product property.
Zero Product Property:
- If the product of two expressions is zero, then one or both of the expressions equal zero.
- If A and B are expressions and AB=0, then A=0 or B=0
- If (x+5)(x+2)=0, then x+5=0 or x+2=0. So x=-5 or x=-2.
Zeros of a Function
The x-intercepts of the graph of y=a(x-p)(x-q) are p and q. Because the function's value is zero when x = p and when x = q, the numbers p and q are called zeros of the functions
You can also use a graphing calculator to help you.