Algebra chapter 8: Quadratic Expressions and Equations

Square of a sum and difference (special products)

- (a+b) (a-b) - = a^2-b^2 Example: (2x^2+3)(2x^2-3) - Use the F.O.I.L Method - 4x^4-6x^2+6x^2-9 - 4x^4-9

Square of a sum (special products)

- (a+b)^2= (a+b)(a+b) = a^2+2ab+b^2 Example: (3x+5)^2 - (3x+5)(3x+5) - 3x^2+2(3x)(5)+5^2 -Multiply, combine like terms, and write in standard form - 9x^2+30x+25

Square of a difference (special products)

- (a-b)^2= (a-b)(a-b) = a^2-2ab-b^2 Example: (2x-5y)^2 - (2x-5y)(2x-5y) - 2x^2-2(2x)(5y)+5y^2 - Multiply, combine like terms, and write in standard form - 4x-20xy+25y^2

Differences of squares

- (a^2-b^2) (a+b) (a-b) - 16h^2-9a^2 - (4h)^2-(3a)^2 - (4h+3a)(4h-3a)

The Box Method (for multiplying polynomials)

- Draw your box. The height of the box should be one more than the degree of the first polynomial. The length of the box should be one more than the degree of the second polynomial. - Write the polynomials you are multiplying on the outside edges of the box. Include ZEROS if necessary. For example, x^2-1 should be x^2+0x-1. - Multiply to get each box's value - like terms can be found on the diagonals of your box. Combine them! - Write your answer in standard form

Factoring polynomials (with a leading coefficient other than 1)

- Is there a GCF? If so, factor it out - Kidnap! (multiply axc) - Factor as if a=1 - Bring back 'a' and find imposter - Pull out imposter by GCF and banish - Check your work! (NOTE: you are still using the big 'X' when you factor)

The F.O.I.L Method (for multiplying binomials)

- Multiply First terms - Multiply Outer terms - Multiply Inner terms - Multiply Last terms Example: (b+3) (b-2) - First terms: bxb= b^2 - Outer terms: bx-2=-2x - Inner terms: 3xb=3x - Last terms: 3x-2=-6 - x^2+-2x+3x+-6 - Combine like terms - x^2+x-6

Multiplying a monomial and polynomial

- Use the distributive property - Multiply the terms - Combine like terms - Write in standard form Example 1: -3x^2(7x^2-x+4) - (-3x^2)(7x^2)+(-3x^2)(-x)+(-3x^2)(4) - -21x^4+3x^3-12x^2 - Already in standard form! Example 2: 3(5x^2+2x-4)-x(7x^2+2x-3) - (3)(5x^2)+(3)(2x)+(3)(-4)+(-x)(7x^2)+(-x)(2x)+(-x)(-3) - 15x^2+6x-12-7x^3-2x+3x - 13x^2+9x-12-7x^3 - -7x^3+13x^2+9x-12 Example 3:2a(5a-2)+3a(2a+6)+8=a(4a+1)+2a(6a-4)+50 - 2a(5a)+2a(-2)+3a(2a)+3a(6)+8= a(4a)+a(1)+2a(6a)+2a(-4)+50 - 10a^2+-4a+6a^2+18a+8=4a^2+a+12a^2-8a+50 - 16a^2+14a+8=16a^2-7a+50 - Cancel terms, and isolate the variable on one side of the equal sign - 21a=42 - a=2

Factor by grouping

- this method of grouping requires an even number of terms. - x(a+b)+y(a+b) - (x+y) (a+b) - (2mk-12m)+(42-7k) - 2m(k-6)+-7(-6+k) - (2m-7)(k-6)

Factoring polynomials (with a leading coefficient of 1)

- x^2+bx+c - Set up parenthesis with your variable in front of each (bxb=b^2) - Set up the big x (X) adn ask what multiplies to 'c' and adds to 'b' - Solve "X" puzzle, watch signs, and put answers in parenthesis - Check your work by the F.O.I.L. method

How to find the standard form of a polynomial?

-Arrange the terms so the term with the highest exponent goes first (Leading Term) -Follow with the other terms with exponents decreasing order -The last term should be the constant Example: 3(2x^2+5)-x^3-2x^2-1 Distribute: 6x^2+15-x^3-2x^2-1 Combine like Terms: 4x^2+14-x^3 Standard Form:-x^3+4x^2+14

Zero Product Property

-If the product of two factors is 0, then at least one of the factors must be 0 - (2d+6)(3d-15)=0 - 2d+6=0 3d-15+0 - Isolate the variable to one side of the equal sign - 2d= -6 3d=15 - d=-3 d=5

Subtracting polynomials

-Write each polynomial in parenthesis -Asks for the difference, you subtract them -Apply the distributive property -Combine like terms -Write in standard form Example: *Find the difference* x^2-5x-4 and 2x^2-11x+9 - (x^2-5x-4)-(2x^2-11x+9) -(Distribute the operation. NOTE: subtraction= opposite sign) (x^2-5x-4)-(-2x^2+11x-9) - x^2 +6x-13 -Already in standard form!

Adding polynomials

-Write each polynomial in parenthesis -Asks for the sum, you add them -Apply the distributive property -Combine like terms -Write in standard form Example: *Find the sum* x^2-5x-4 and 2x^2-11x+9 - (x^2-5x-4)+(2x^2-11x+9) -(Distribute the operation. NOTE: addition= same sign) (x^2-5x-4)+(2x^2-11x+9) - 3x^2-16x+5 -Already in standard form!

Polynomial

A monomial, or the sum of monomials

Terms

A part of a polynomial that defines whether it is a monomial, binomial, trinomial, or polynomial. they are always being added or subtracted in the polynomial Example: -5x+3x^2-12+x^3 Terms: -5x, 3x^2, x^3

Prime polynomial

A polynomial that cannot be written as a product of two polynomials with integral coefficients

Standard form of a polynomial

A polynomial that is written with the terms in order from greatest degree to least degree.

Square root property

A quadratic equation involving a single variable term to the second power can be solved by taking the square root of both sides of an equation. NOTE: when you take the square root of a whole number, you must add a positive and negative sign

Perfect square trinomial

A trinomial that is the square of a binomial

Quadratic equation

An equation that can be written in the form ax^2 + bx + c = 0, where a is not zero.

Quadratic expression

An expression in one variable with a degree of 2 written in the form of ax^2+bx+c

Factoring out the GCF (Greatest Common Factor)

Factoring simply means to undistribute make sure you always factor out the Greatest Common Factor. Example: 27y^2+18y - Find the GCF - 9y(27y^2+18y) - 9y(3y+2)

Coefficients

If a term has a variable, the number in front of the variable is the coefficient. NOTE: Each coefficient takes the sign in front of it. If the coefficient is not written, it is an invisible 1. Example: -5x+3x^2-12+x^3 Coefficients: -5, 3, 1

Constants

If a term has no variables it is a Constant. NOTE: The constant takes the sign in front of it. Example: -5x+3x^2-12+x^3 Constant: -12

Zero product property

If the product of two factors is zero, then at least one of the factors must be zero. Ex: If ab = 0, then a = 0 or b = 0.

Degree

If your polynomial only has one variable, the gegree og the polynomial is the exponent of the leading term (term with the highest variable) Example: -5x+3x^2-12+x^3 Degree: 3rd degree

0 degree of a polynomial

It has a degree of 0, it is known as Constant. Example: 3

1st degree of a polynomial

It has a degree of 1, it is known as Linear. Example: 2x+1

2nd degree of a polynomial

It has a degree of 2, it is known as Quadratic. Example: x^2-4x

3rd degree of a polynomial

It has a degree of 3, it is known as Cubic. Example: y^3+y^2-4

4th degree of a polynomial

It has a degree of 4, it is known as Quartic. Example: x^4-3

5th degree polynomial

It has a degree of 5, it is known as Quintic. Example: y^5+y^2

6th degree (or higher) polynomial

It has a degree of 6 or higher, it is known as a 6th (or 7th, 8th....) degree polynomial. Example: z^6-z^3

How to find terms?

Terms are always separated by addition or subtraction, NOT multiplication or division

Leading Coefficient

The Coefficient of the Leading Term (the term with the highest exponent) is known as the Leading Coefficient. Example: -5x+3x^2-12+x^3 Leading Coefficient: 1

Leading coefficients

The coefficient of the term with the highest degree in a polynomial

Degree of a polynomial

The greatest degree of any term in the polynomial

How to find the degree of a polynomial?

The highest exponent is the same as your degree. If the highest exponent is 4, you have a 4th degree polynomial. NOTE: for a polynomial with only one variable, the degree is the largest exponent of that variable.

Degree of a monomial

The sum of the exponents of all of the variables in a monomial

Trinomial

The sum of three monomials, and a type of polynomial

Binomial

The sum of two monomials, and a type of polynomial

Leading Term

The term that has the highest exponent is known as the Leading Term Example: -5x+3x^2-12+x^3 Leading Term: x^3

Factoring by group

The use of the Distributive Property to factor some polynomials having four or more terms.

Factoring

To express a polynomial as the product of monomials and polynomials

F.O.I.L method (First, Outer, Inner, Last)

To multiply two binomials, find the sum of the products of the First terms, the Outer terms, the Inner terms, and the Last terms

Difference of two squares

Two perfect squares that are separated by a subtraction sign like this: a^2 - b^2

How to name polynomials?

combine the degree and more specific group of polynomials. Example: -7+3n^3 is a Cubic Binomial 5 is a Constant Monomial -x^4+3x^2-11 is a Quartic Trinomial