POLYNOMIALS

 TUNE: "HOKEY POKEY" You put your right hand out, You put your left hand out, You can even flip direction Or jump straight up and down.**  **Now** "Polynomials can dance about!" You put your right hand up, You put your left hand down, You can alternate direction Or jump straight up and down. **  **Now** "Polynomials can dance about!" You put your right hand up, You put your left hand up, You can even flip direction Or jump straight up and down. Now that's a quadratic function And it makes you want to shout, "Polynomials can dance about!" You put your right hand up, You put your left hand down, You can alternate direction Or jump straight up and down. **  **Now** ** And it makes you want to shout, "Polynomials can dance about!"
 * THE POLYNOMIAL DANCE**
 * WORDS BY: DANE R. CAMP
 * that's a constant function **
 * And it makes you want to shout,
 * that's a linear function **
 * And it makes you want to shout,
 * that's a cubic function **

Added to this site by Connor Merritt

POLYNOMIAL VOCABULARY:** F UNCTIONS CAN BE CATEGORIZED, and the simplest type is a polynomial function. We will define it below. We begin with vocabulary.  1. When numbers are added or subtracted, they are called terms. This -- 4//x//² + 7//x// − 8 -- is a sum of three terms. (In algebra, we speak of a "sum," even though a term may be subtracted.) When numbers are multiplied, they are called factors. This -- 1. (//x// + 1)(//x// + 2)(//x// + 3) -- is a product of three factors.  2. A variable is a symbol that takes on values. A value is a number. Thus if //x// is the variable and has the value 4, then 5//x// + 1 has the value 21. 3. A constant is a symbol that has a single value. Example 1. The symbols '5' and '' are constants. The beginning letters of the alphabet //a, b, c//, etc. are typically used to denote constants, while the letters //x//, //y//, //z//, are typically used to denote variables. For example, if we write //y// = //a//x² + //b//x + //c//, we mean that //a//, //b//, //c// are constants (i.e. fixed numbers), and that //x// and //y// are variables.  4. A monomial in //x// is a single term of the form //ax// //n//, where //a// is a [|real number] and //n// is a [|whole number]. The following are monomials in //x//: 5//x// 3, −6.3//x//, 2. We say that the number 2 is a monomial in //x//, because 2 = 2//x// 0 = 2**·** 1. ([|Lesson 21] of Algebra.)

5. A polynomial in //x// is a sum of [|monomials] in //x//. The variable of the polynomial, in this case //x//, is also called the argument of the polynomial. Here is a polynomial with argument //t// : //t// ² −5//t// + 1. When we write a polynomial, the style is to begin with the highest exponent and go to the lowest. 4, 3, 2, 1. (For the general form of a polynomial, see [|Problem 6] below.)  6. The degree of a term is the sum of the exponents of all the variables in that term. In functions of a single variable, such as //x//, the degree of a term is simply the exponent. //x// 4, //x//³//y//, //x//²//y//², //x////y//³, //y// 4. In each term, the sum of the exponents is 4. As the exponent of //x// decreases, the exponent of //y// increases. Problem 1. Write all possible terms of the 5th degree in the variables //x// and //y//. To see the answer, pass your mouse over the colored area. To cover the answer again, click "Refresh" ("Reload"). //x// 5, //x// 4 //y//, //x// 3 //y// 2 , //x// 2 //y// 3 , //x////y// 4 , //y// 5.  7. The leading term of a polynomial is the term of highest [|degree]. 8. The leading coefficient of a polynomial is the coefficient of the leading term.  9. The degree of a polynomial is the degree of the [|leading term]. Here is a polynomial of the first degree: //x// − 2. 1 is the highest exponent.  10. The constant term of a polynomial is the term of degree 0; it is the term in which the variable does not appear. Problem 2. Which of the following is a polynomial? If an expression is a polynomial, name its degree, and say the variable that the polynomial is in. a) //x// 3 − 2//x//² − 3//x// − 4 Polynomial of the 3rd degree in //x//. b) 3//y//² + 2//y// + 1 Polynomial of the 2nd degree in //y//. c) //x// 3 + 2 + 1 This is not a polynomial, because is not a whole number power. It is //x// ½ . d) //z// + 2 Polynomial of the first degree in //z//. //x// || //x// || = //x// −1, which is not a whole || number power. Problem 3. Name the degree, the leading coefficient, and the constant term. a) //f//(//x//) = 6//x// 3 + 7//x//² − 3//x// + 1 3rd degree. Leading coefficient, 6. Constant term, 1. b) //g//(//x//) = −//x// + 2 1st degree. Leading coefficient, −1. Constant term, 2. c) //h//(//x//) = 4//x// 5 5th degree. Leading coefficient, 4. Constant term, 0. d) //f//(//h//) = //h//² − 7//h// − 5 2nd degree. Leading coefficient, 1. Constant term, −5.  If we were to multiply out, then the degree of the product would be the sum of the degrees of each factor: 1 + 1 + 3 = 5. For, (5//x// + 1)(3//x// − 1)(2//x// + 5)³ = (5//x// + 1)(3//x// − 1)(2//x// + 5)(2//x// + 5)(2//x// + 5). The leading coefficient would be the product of all the leading coefficients: 5**·** 3**·** 2³ = 15**·** 8 = 120. And the constant term would be the product of all the constant terms: 1**·** (−1)**·** 5³ = −1**·** 125 = −125. Problem 4. Name the degree, the leading coefficient, and the constant term. a) //f//(//x//) = (//x// − 1)(//x//² + //x// − 6) Degree: 3. Leading coefficient: 1. Constant term: 6. b) //g//(//x//) = (//x// + 2)²(//x// − 3) 3 (2//x// + 1) 4 Degree: 9. Leading coefficient: 1²**·** 1³**·** 2 4 = 16. Constant term: 2²**·** (-3)³**·** 1 4 =4**·** (−27)= −108 c) //f//(//x//) = (2//x// + 1) 5 Degree: 5. Leading coefficient: 2 5 =32. Constant term: 1 5 = 1. d) //h//(//x//) = //x//(//x// − 2) 5 (//x// + 3)² Degree: 8. Leading coefficient: 1. Constant term: 0.  11. The general form of a polynomial shows the terms of all possible degree. Here, for example, is the general form of a polynomial of the third degree: //a////x//³ + //bx//² + //cx// + //d// Notice that there are four constants: //a//, //b//, //c//, //d//. In the general form, the number of constants, because of the term of degree 0, is always one more than the degree of the polynomial. Now, to indicate a polynomial of the 50th degree, we cannot indicate the constants by resorting to different letters. Instead, we use sub-script notation. We use one letter, such as //a//, and indicate different constants by means of sub-scripts. Thus, //a// 1 ("//a// sub-1") will be one constant. //a// 2 ("//a// sub-2") will be another. And so on. Here, then, is the general form of a polynomial of the 50th degree: //a// 50 //x// 50 + //a// 49 //x// 49 +. . . + //a// 2 //x// 2 + //a// 1 //x// + //a// 0 The constant //a// //k// -- for each sub-script //k// (//k// = 0, 1, 2, . . ., 50) -- is the coefficient of //x// //k//. Notice that there are __51__ constants. The constant term //a// 0 is the 51st. Problem 5. a) Using subscript notation, write the general form of a polynomial of a) the fifth degree in //x//. //a// 5 //x// 5 + //a// 4 //x// 4 + //a// 3 //x// 3 + //a// 2 //x// 2 + //a// 1 //x// + //a// 0 b) In that general form, how many constants are there? 6 c) Name the six constants of this fifth degree polynomial: //x// 5 + 6//x//² − //x.// a 5 =1. a 4 = 0. a 3 =0. a 2 = 6. a 1 =−1. a 0 = 0.
 * Example 2.** 5//x// 3 − 4//x//² + 7//x// − 8.
 * Example 3.** The term 5//x// 3 is of degree 3 in the variable //x//.
 * Example 4.** This term 2//x////y// 2 //z// 3 is of degree 1 + 2 + 3 = 6 in the variables //x//, //y//, and //z//.
 * Example 5.** Here are all possible terms of the 4th degree in the variables //x// and //y//:
 * Example 6.** The leading term of this polynomial 5//x//³ − 4//x//² + 7//x// − 8 is 5//x//³.
 * Example 7.** The leading coefficient of that polynomial is 5.
 * Example 8.** The degree of this polynomial 5//x//³ − 4//x//² + 7//x// − 8 is 3.
 * Example 9.** The constant term of this polynomial 5//x//³ − 4//x//² + 7//x// − 8 is −8.
 * e) //x//² − 2//x// + || __1__
 * Not a polynomial, because || __1__
 * Example 10.** Name the degree, the leading coefficient, and the constant term of (5//x// + 1)(3//x// − 1)(2//x// + 5)³.

ADDITION AND SUBTRACTION OF POLYNOMIALS:This link will help to understand the steps of adding and subtracting polynomials. [|adding and subtracting polynomials] Here is an example of a polynomials off of the Mock SOL which is similar to the real SOL- (2x^2 – 5x + 6) + ( 5x^2 – 3x + 4) The answer choices were-

A) 7x^2 – 8x +10 B)7x^2 – 2x + 10 C)7x^2 – 8x + 2 D)7x^2 – 2x + 2 The correct answer is A. You would add the X values but not the exponents. The exponents stay the same. Then you would add the -3x with the -5x to get 8x. The the final step would be to add the 6 and the 4 to get 10.

Adding]] and Subtracting Polynomials The most common process when doing polynomials is combining the like terms. The goal is to add as many like terms into one. You need to be careful and watch what you do because you can't add and subtract certain things. Terms can only be combined if they have the exact same variable part. For example 4x and 3 aren't like terms because the 3 doesn't have X as a variable. 4x and 3x are like terms so they can be combined. You have to be sure not to combine terms with exponents like for example 4x^2 + 3x. You can't do this because one has a exponent and the other doesn't. If there was (4x^2) + (6x^2) you could add the terms because they have equal exponents. The answer for this would be 10x^4. This problem written out is 4xx + 6xx =10xxxx. The number x's represent what the exponent is. Also different exponents can't be added or subtracted. This link will provide extra help. []

__MULTIPLICATION AND DIVISION OF POLYNOMIALS__: Multiplying polynomials is a long but easy process.The following is a example: (x+3)(3x^2+2x+5) -The parentheses show that its multiplication, because nothing is in between them. (x)(3x^2)+(x)(2x)+(x)(5)+(3)(3x2)+(3)(2x)+(3)(5) - You use the distributive property to multiply the terms in the first parentheses with all of the ones in the next parentheses individually. 3x^3+2x^2+5x+9x^2+6x+15 - Now, you do the multiplications you just rearranged. 3x^3+11x^2+11x+15 - Now combine like terms. The problem is now complete.

__Division__ Division is just as simple as multiplication.Below is a example: __(7x^3+9x^2+15x)__ - This is our problem. You have to divide everything thats on top by whats on the bottom. *2x __7x^3__ + __9x^2__ + __15x__ - We have separated it, so do x zero pairs *2x *+ 2x + 2x __7x^2__ + __9x^2__ + __15__ - This is all we have left, and its the answer. 2** 2***** 2 ^=exponent
 * =buffer for fractions