Factoring Trinomials

In the last few blogs we have talked about multiplying binomials to produce a trinomial.

Now we are going to practice undoing or FACTORING trinomials.

We begin by listing all the factors of the constant term (term with no variable).

We look at the factors and see if any of them will add up to the middle term.










Last we look at the signs in the trinomial.  The last sign tells us whether the signs in the binomals are the same or different.  

The only way to multiply two numbers and get a positive number is for both numbers to be positive or both number are negative.

The sign for the middle term tells us if the signs in the binomials are poistive or negative.

When both signs in the trinomial are positive the signs in the parenthesis are positive.

(x + 3)(x + 10)

Mulitpying Special Binomials

When we multiply binomials there are times we run across special cases.

Most binomial multiplication looks like:

(x + 4)(x - 3)   or   (x - 2)(x - 7)    or   (x + 4)(x + 4)

When we multiply binomials with different constants (the number on the end without a variable), we end up with x squared, two like terms and a constant.

 We combine the like terms and end up with a trinomial written in standard form.



Squaring Binomials



When you square a binomial you must expand it to multiply each binomal using the FOIL method.









Difference of Two Squares

The difference of two squares is the product of the sum and difference of the same two terms and equals the difference of their squares.

Notice there is no middle term with the difference of two squares!

Even if we have more complicated binomials we still end up to the difference of perfect squares and no middle term.

FOIL Method

When we multiply binomials we use distribution. One way to remember the sequence for multiplying each term by each term is the FOIL method.


FOIL

F = first terms in each parenthesis

O = outer two terms in each parenthesis

I = inner two terms in the parenthesis

L = last two terms in the parenthesis











Combine like terms



Show your final answer

Polynomials

Nomial is another name for number or term. Poly means many. A polynomial is many numbers or many terms.

In algebra 1 we work with:

Monomials - - which means one term

Examples:

5
x
12y
-8xyz

Binomials - - which means two terms

Examples:

2x - 6
x + y
-8xyz + 12x

Notice that binomials are in lowest terms and have no like terms that can be combined.


Trinomials - - which means three terms

Examples:















Each of these polynomials also has a DEGREE (or power). We find the degree by finding the highest power of each term. The term with the highest power sets the degree for the polynomial.


Polynomials can be:
constant
(0 degree)

linear
(1st degree)

quadratic
(2nd degree)

cubic (3rd degree)

4th degree

5th degree

and on and on.






However, the highest degree of a polynomial is the combination of the powers in each individual term.








Standard Form of a Polynomial

As we have learned this year, there are specific ways to write algebraic terms. When writing polynomials we write them from the highest exponent to the lowest exponent.




They are also written in reduced form so all mathematical process that can be done, must be done and all like-terms must be combined so the polynomial is in standard form.


Your final answer must be written in standard form, in lowest term.




Adding Polynomials



Adding vertically may be the easiest procedure:












Remember to line up like terms so you are combining like variable with the same exponent.










Subtracting Polynomials


You must remember to distrbute the negative sign into the parentheses.

Graphing Exponential Functions

Graphing exponential functions can be as easy as graphing a linear function.


This is an exponential function. It is exponential because x is the exponent of 2.








To graph this equation you can make a t-table of values for x and calculate the value for y.













Then you graph the point for (x, y)


You will get half of a parabola.

Graphing Inequalities

We have already graphed inequalities on a number line but what is the procedure and outcome of graphing inequalities on a coordinate plane?

As with number line graphs, we need to simplify the inequality expression before we try to graph.


Therefor: 2y > 4x - 6 must be simplified in the same manner as an equaiton.

We need y to have a coefficient of 1 so would divide each term in the inequality by 2.

2y > 4x - 6
2


y > 2x - 3

We would then graph the inequality the same way we graph equations, beginning with the y-intercept (-3) and then using the slope to extend the line in two directions.

Negative Exponents

Negative exponents have nothing to do with negative numbers.

If we think of the use of negative exponents in Scientific Notation we can understand them a bit better.



A negative exponent with a power of 10 means it is a decimal.




10 to the -3 = 0.001, which we can also write as 1 over 1000.





When we use powers of 10 in Scientific Notation we are changing the negative power to a decimal and multiplying it by the significant digits.




Significant digits = 2.63

10 to -3 = 0.001

Multiply and you have 0.00263









A negative exponent means "the reciprocal"


2 to the -3 can be rewritten as 1 over 2 to the 3rd (or 2 cubed).

2 cubed is 8.

Our final answer is one over eight.





When we are simplifying a more complex problem that includes a negative exponent we:





reduce any fractions

combine the exponets for each base:

x
and
y

Finally put any negative exponents in the opposite area of the fraction:


If there is a negative in the numerartor -> put it in the denominator.

If there is a negative in the denominator -> put it in the numerator.

Your final answer should never include a negative exponent.