2. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI Monomials, binomials, and Polynomials Monomials, binomials, and polynomials are the names used quite often in Algebra. They simply denote the number of terms used in any given formula or equation. For example: Monomial simply mean a one- term expression such as X 2, 4x 3, a, 2x Etc. Binomial mean a two-term expression such as 3x + y or x 2 – 6 Etc. Polynomials represent expressions that have more than 2 terms. Although the expression with 3 terms is sometimes called trinomial, in most cases it is simply called a Polynomial. Here are some examples of Polynomials 2x + y + z x 4 + 4x 2 – 4 (2a + b) + 2c 2 2x 3 – x 2 – 4x + 2 In addition, polynomials have certain rules that defines it as a Polynomial. It is very important to look these rules in more detail:

3. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI Polynomial terms have variables which are raised to whole-number power ONLY; there are no square roots of variables, no fractional powers, and no variables in the denominator of any fractions. Here are some examples: 2x –3 This is NOT a polynomial term because the exponent (-3) is negative 1/x21/x2 This is NOT a polynomial term because the variable is in the denominator (the lower part of the fraction. x 3 This is NOT a polynomial term because the variable is inside the radical (under the root symbol). 4x24x2 This is a polynomial term because the rules are met and therefore it would be considered a valid polynomial term. FINALLY !!! So, as you can see the rules are quite tough, but there are reasons for all these restrictions. It makes it possible to work with these terms and solve many scientific and technical problems.

4. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI 6x 2 + 3x - 5 Polynomial terminology: Terms Leading term First term of the expression Constant term This is simply a number without any variables Degrees of polynomials: 4x 5 – 2x 3 – 7x + 9 This polynomials has 4 terms, which consist of the following: a fifth degree term a third degree term a first degree term and a constant term. As you can see the degree simply points to the exponents (i.e. the term raised to the 4-th power would be considered a fourth degree term). You can also call the entire expression as a fifth degree polynomial because the largest exponent is 5

5. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI 12x 4 + 7x 2 + x This polynomial has three terms, a fourth-degree term, a second- degree term, and a first-degree term. There is no constant term. This is a fourth-degree polynomial. x 5 - 4x 3 - 2x + 6 8x 7 - 3x 4 + 2x 2 + 14 (4x 2 - 3x 3 ) · 2x + 5 Try to figure out the degrees and term numbers in the three additional examples below: Variables and coefficients: 6x - 4x 3 - 2x + 2 Coefficients Variables Constant Please note, in the example above, 6 would be called a Leading coefficient

6. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI Combining, adding and subtracting polynomials: In many cases it is very important to properly group polynomials to simplify, factor, or simply to arrange factors for easier solutions to many complex problems. This is typically achieved by combining like terms. Consider the following example: x 4 + 4x 2 – 4 + 2x 4 + x +2x + 5 + 3x 4 The best way to do this is to arrange in the order power degrees first (i.e. fifth, fourth, third, etc.) then the constant terms (just numbers, without the variables). In this particular case we will arrange as follows: X 4 + 2x 4 + 3x 4 (all fourth degree) we can add these together to simplify the expression: 6x 4 4x 2 here we only have one term with the second power. We just leave it as it is x + 2x here we only have two terms with the fist degree power. Adding them together will get us 3x -4 + 5 These are the 2 constant terms. Adding these together will give us a 1 So, the final simplified and combined expression will give us 6x 4 + 4x 2 + 3x +1

7. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI 2x 2 + x 2 – 1 + 2x 2 + x + 2x 2 + 6 + 3x = = 7x 2 + 4x + 5 Additional 8 examples: 4y 2 + y 2 + 4 + 5y 2 + y 2 + 3y 2 + 8 + 3y 2 = = 17y 2 + 12 2xy + x + 5 + 5xy + xy + 4xy – 4 = = 12xy + 1 5ab 3 + 2x + 15ab 3 + 5x + ab + 3ab + 4x = = 20ab 3 + 4ab + 11x 5x 2 + x 2 – 1 + 2x 2 + 2x 2 + 6 + 3x – 9 = = 10x 2 + 3x – 4 (note that - 4 was obtained by (-1 +6 -9) = - 4) 7a 5 + y 4 + 4a 3 + 5y 4 + 2a 5 + 3y 4 + 3y 4 = = 9a 5 + 12y 4 + 4a 3 25 – x + 5 – 12x + 10 + 4x – 4 = = -9x - 36 2ab + ab + 5ab + 3ab + ab + 4ab = = 16ab

8. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI Multiplying polynomials: When multiplying polynomials, it is very important to properly treat the exponents as well as coefficients and variables. Consider the following examples: (x + 3) · 2 = (x · 2) + (3 · 2) = 2x + 6 Here we have to multiply each term within the parenthesis by 2 So, x · 2 = 2x and 3 · 2 = 6 That gave us the final answer of 2x + 6 (2x + 4) · 3 = (2x · 3) + (4 · 3) = 6x + 12 (5x - 4) · 4 = (5x · 4) - (4 · 4) = 20x - 16 Additional 2 examples: When multiplying 2 groups of polynomials within the parenthesis, similar rules apply, but it is a little bit more involved. Please carefully examine the following example: (x + 3) · (x + 2) this could also be written without the multiplication dot – it would mean exactly the same thing (x + 3)(x + 2) = = (x · x) + (x · 2) + (3 · x) + (3 · 2) = X 2 + 2x + 3X + 6 = X 2 + 5x + 6 Here we had to multiply every term in the first parenthesis by every term in the second parenthesis. It may be a little confusing in the beginning, but after couple of times it becomes very easy. Please see a more detailed explanation on the next page:

9. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI (x + 3)(x + 2) Again, here is the problem from the previous page: First, we multiplied x in the first parenthesis by x in the second parenthesis. That gave us an x 2 Then, we multiplied x in the first parenthesis by 2 in the second parenthesis. That gave us an 2x Next, we multiplied 3 in the first parenthesis by x in the second parenthesis. That gave us an 3x And finally, we multiplied 3 in the first parenthesis by 2 in the second parenthesis. That gave us an 6 At the end we simply added the results: X 2 + 2x + 3x + 6 Of course we have to add the 2x + 3x which gave us the 5x So that the final answer was X 2 + 5x + 6 This is the final answer, it could not be simplified any further

10. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI Polynomial Factorization: Factorization is used quite often in algebra. It helps to organize and in many cases to simplify and solve some of the more complex scientific and technical problems. Here are some very important factorization formulas to remember. Each formula starts by showing the end result and then how it was derived: ac + ad = a(c + d) ac + bc + ad + bd = (a + b)(c + d) a 2 – b 2 = (a+b)(a-b) (a + b) 2 = a 2 + 2ab + b 2 (a - b) 2 = a 2 - 2ab + b 2 (a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3 (a - b) 3 = a 3 - 3a 2 b + 3ab 2 - b 3 a 3 – b 3 = (a - b)(a 2 +ab + b 2 ) a 3 + b 3 = (a + b)(a 2 - ab + b 2 ) a 4 – b 4 = (a - b)(a 3 + a 2 b + ab 2 + b 3 ) a 5 – b 5 = = (a - b)(a 4 +a 3 b + a 2 b 2 + ab 3 b 4 ) a 6 – b 6 = = (a - b)(a 5 +a 4 b + a 3 b 2 + a 2 b 3 + ab 4 + b 5 ) Remember these formulas may not be applied every time, but they a commonly used in Algebra and should be remembered.

11. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI Introduction to Equations: An equation, in Mathematics, generally means a statement that shows the equality of two expressions. In Algebraic notation, this is written by placing the expressions on either side of an equal sign(=). For example: 5x + 10 = 20 Expression 1 Expression 2 2x + 5 = 3x + 2 Expression 1 Expression 2 Sometimes it comes in this form: But in many cases equations come in a special form, where the second expression is simply zero (0). It maybe hard to visualize at first, but as you will see in the next section, these equations are quite useful, not only in Mathematics, but in real life as well. These equations can solve many real-life problems in science, engineering, and architecture. 4x + 2 = 0