Definition binomial and polynomials
Definition polynomial: A polynomial in x is a term or the sum of a finite
number of terms of the form axn; for any real a and any whole number n
The polynomial is written in descending powers because the exponents
decrease from left to right.
Three type of polynomials are very common and are given special names
1) A polynomial with only one term is a monomial
2) A polynomial with two terms is a binomial.
3) A polynomial with three terms is a trinomial
On the other hand :the degree of a term is the sum of exponents of the variable.
A constant has degree 0 const =17=17x0 and any number to the zero power
is 1. 17x0 = 17(1)=17
Note 3x4has degree 4,while 6x17 has degree 17. Note 2 : 2x2y has degree
of 2+1=3 from x2y1.
Rule any number or variable raised to the power of 1 is the number or variable itself. x1=x ,y1=y
Evaluating a polynomial
Find the value of 3x4+5x3-4x-4
when Case 1 x=-2 and when Case 2 x=3. Note evaluate only variable at a time.
Case 1: 3x4+5x3-4x-4 = 3(-2)4+5(-2)3-4(-2)-4 = 12
Case 2: 3x4+5x3-4x-4= 3(3)4+5(3)3-4(3)-4 = 362
Definition: whole number(positive integer including 0). Note real numbers are all numbers on the horizontal number line as opposed to the numbers on the vertical (imaginary axis) i (a) =sqrt -1(a)
For example
Real numbers and the number line
Whole numbers 0,6,15
Integers -11,-12, 0,6,15
Rational numbers
Irrational Numbers
foilFOIL
Multiplying binomials by the FOIL Method
In algebra, many of the polynomials to be multiplied are both binomials (with just two terms). For these products the FOIL Method reduces
the rectangle method to a systematic approach without the rectangles.
To develop the FOIL methods we use the distributive property to find
(x+3)(x+5)
(x+3)(x+5)= (x+3) x+ (x+3)5
x(x) +3(x) +x (5) +3(5)
=x2+3x+5x+15
=x2+8x+15
(x+3)(x+5)
FOIL
Multiplying binomials by the FOIL Method
In algebra, many of the polynomials to be multiplied are both binomials (with just two terms). For these products the FOIL Method reduces
the rectangle method to a systematic approach without the rectangles.
To develop the FOIL methods we use the distributive property to find
(x+3)(x+5)
(x+3)(x+5)= (x+3) x+ (x+3)5
x(x) +3(x) +x (5) +3(5)
=x2+3x+5x+15
=x2+8x+15
(x+3)(x+5) Multiply the First two terms: x(x) F
^ ^
| |
(x+3)(x+5) Multiply the Outer terms: x (5) O
^ ^
| … |
(x+3)(x+5) Multiply the Inner terms: 3(x) I
^ ^
| |
(x+3)(x+5) Multiply the Last terms: x (5) L
^ ^
|… |
The inner product and the outer product should be added mentally
so that the three terms can be written without the extra step.
Multiplying Binomials by the FOIL method
Step 1 Multiply the First terms of the binomial to get the first terms of the answer.
Step 2 Find the Outer product and the Inner produce and add them (when possible) to get the middle term of the answer.
Step 3 Multiply the two last terms of the answer to get the term of the answer.
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