Intro to Polynomials

Factor the following expression.

1. 5x + 25

Step 1: Find the greatest common factor (GCF). The largest number that divisible by both number is 5. Therefore, your GCF is 5.

Step 2: Factor (break down) your expression into a simpler form. In this case, your factored form is as follows:
5(x + 5).

 

Find the sum or difference of the following polynomial.

(x^3 + 2x + 1) + ( 3x^2 + 4 )

Step 1: If an addition sign is presented, you must find the sum of the polynomial. If a subtraction sign is presented, you must find the difference of the polynomial. In this case, you will find the sum.

Step 2: Combine like terms. Remember, in addition, you cannot combine exponents. All exponents should be written in descending (5, 4, 3,)  order. The only two terms that can be combined in this polynomial is 1 and 4, which will yield a result of 5.

Step 3: Rewrite your expression. Your final answer is x^3 + 3x^2 + 2x + 5.

 

Multiply the following binomial.

When you are multiplying binomials, you can use two methods. The first method is called the FOIL method. FOIL is an acronym for First Outside Inside Last. Another version of FOIL is called the BOX method.

1. (x + 3) (x + 8)            Step 1: Multiply the first terms in both parentheses (x * x = x^2)

Step 2: Multiply the first term in the first parentheses and last term in the second parentheses                                                                     (x * 8 = 8x)

Step 3: Multiply the last term in the first parentheses and the first term in the second parentheses                                                             (3 * x = 3x)

Step 4: Multiply the last terms in both parentheses ( 3 * 8 = 24)

Step 5: Combine the result of Step 2 & 3.

Step 6: Rewrite your final answer.

Your final answer is x^2 + 11x + 24.

 

Find the area of the garden.

Thomasina wants to plant a rectangular rose garden that measures (x + 5) by (6x + 7). What is the area of Thomasina’s rose garden?

Step 1: Write the formula for finding the area of a rectangle. {A = l x w}

Step 2: Substitute your length and width into the formula. {A= (x+5) (6x+7) }

Step 3: Use the FOIL method to multiply the binomial. { A = x^2 +30x + 7x + 35}

Step 4: Combine like terms.  { 30x + 7x = 37x)

Step 5: Write your final answer. { A = x^2 + 37x + 35 }

Factoring:Greatest Common Factor

There is a factoring that is easier than FOIL and that is called the GCF or the Greatest Common Factor. The greatest common factor is a number that divisible by a set of numbers. For instance, the gcf of 12 and 18 is 6 because 12/6 = 2 and 18/6 = 3.

Well, what if there isn’t any coefficients? What if there is only variables and exponents? The same rules apply. For example, if I were to find the greatest common factor between x^3 and x^2, I must factor each term and eliminate my Xs.

ex.  x^3=  x* x* x    and x^2 = x * x

eliminate  x*x*x and x*x

The greatest common factor between these two terms is x^2.

Remember the factor BOX that Ms. Umeh showed in class?

First box: first term

Second box: second term

Third box: third term

If your middle term is a negative, your operations when you factor must also be negative. If your last term is negative, then your operations must be addition and subtraction.

 

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Function

Answer this question

What do you believe is the most difficult concept that you have learned in Algebra 1?

Graphing

Factoring

Functions

Absolute Value

FOIL/The Box

Perfect Squares

There are two types of polynomials: Perfect Squares and Difference of Squares. You can express either type of polynomial in its general form or factored form. An example of each is listed below.

PERFECT SQUARE

General form:  

Factored form:

DIFFERENCE OF SQUARE

General form:  

Factored form: (m-n)(m+n)

DETERMINING WHETHER A TRINOMIAL IS A PERFECT SQUARE

A trinomial consists of three terms. In order to determine whether a trinomial is a perfect square, simplify each term by its square root.

ex.

REMEMBER: The 1st term and the 3rd term must be perfect squares. The second term is the product of the first and third term multiplied by 2.

First term:  

Third term:

Second term:  2(x)(4)=8x

How do you know if your trinomial is a perfect square? If the product of your second term is the same as the second term listed in the trinomial, then you have a perfect square. The example listed above is  a perfect square.

Greatest Common Factor (GCF) and FOIL

Greatest Common Factor (GCF): The largest number that is evenly divisible between a set of numbers.

FOIL: First, Outside, Inside, Last

Multiply the first terms within each parentheses. Then add/subtract  the numbers/variables on the outside of the parentheses. Follow that by adding/subtracting the numbers/variables on the inside of the parentheses.                Finally, multiply the last two terms in the parentheses.

 

smiling man

 

(n + 2) (n + 6)

n (n) = n squared |  2 (n) = 2n |  6(n) = 6n | 6(2) = 12

Final:

 

GCF: 3

Factored:

Find the product.

When you find the product of a number, it means that you are multiplying the numbers/variables together.

2x(-2x – 3) = 4x squared – 6x