Transcript 7.6 Polynomials and Factoring (1)

7.6 Polynomials and Factoring
Part 1: Polynomials
Basic Terminology
• A term, or monomial, is a number, a variable, or a product of
numbers and variables.
• A polynomial is a term or a finite sum/difference of terms,
with only nonnegative integer exponents on the variables.
– A polynomial CANNOT have a variable in a denominator.
– A polynomial with exactly two terms is a binomial; one
with exactly three terms is a trinomial.
• The greatest exponent in a polynomial in one variable is the
degree of the polynomial.
– The degree of a polynomial in more than one variable is
equal to the greatest degree of any term appearing in the
polynomial.
Examples of Various Polynomials
Type
Monomial
Binomial
Trinomial
Polynomial
Example
-10r6s8
29x11 + 8x15
9p7 – 4p3 + 8p2
5a3b7 – 3a5b5 + 4a2b9 – a10
Degree
14
15
7
11
Addition and Subtraction
• Like terms are terms that have the exact same
variable factors.
• Polynomials are added by adding coefficients
of like terms.
• Polynomials are subtracted by subtracting
coefficients of like terms.
• Polynomials in one variable are often written
with their terms in descending powers; so the
term of the greatest degree is first, and so on.
Adding and Subtracting
Polynomials
• Add or subtract, as indicated.
(2y4 – 3y2 + y) + (4y4 + 7y2 + 6y)
(-3m3 – 8m2 + 4) – (m3 + 7m2 – 3)
Add or subtract, as indicated.
8m4p5 – 9m3p5 + (11m4p5 + 15m3p5)
4(x2 – 3x + 7) – 5(2x2 – 8x – 4)
Multiplication
• Polynomials can be multiplied “horizontally”
using the Distributive Property.
• They can also by multiplied “vertically”.
• Both methods work – it’s your choice! 
Multiplying Polynomials Vertically
• Multiply (3p2 – 4p + 1)(p3 + 2p – 8).
Using the FOIL Method
(Multiplying Horizontally)
• The FOIL method is a convenient way to find
the product of two binomials.
F – first
O – outer
I – inner
L – last
• To multiply other polynomials, use the
Distributive Property in a similar way.
Using the FOIL Method
• Find the product.
(6m + 1)(4m – 3)
Find the product using the
method of your choice.
(2x + 7)(2x – 7)
Special Products
• The last example demonstrated a “special product”
known as the difference of two squares.
Product of the Sum and Difference of Two Terms
(x + y)(x – y) = x2 – y2
Squares of Binomials
(x + y)2 = x2 + 2xy + y2
(x – y)2 = x2 – 2xy + y2
These are just “short
cuts” – If you forget,
just multiply it out!
Finding Special Products
• Find each product.
(3p + 11)(3p – 11)
(2m + 5)2
Find each product.
(5m3 + 3)(5m3 – 3)
(3x – 7y4)2
(9k – 11r3)(9k + 11r3)