Polynomials-Target+B-Polynomial+Operations-Guided+Learning

Polynomials Target B: Perform operations on polynomials.
2x 2  + 6x 2x . x + x + x + x + x +x +x x(2x) + 3x + 3x 2x 2  + 3x 2x + 6
 * [[image:Discuss this.png width="208" height="117"]] || Which of the following expressions are equivalent to 2x(x + 3)? Justify your reasoning.

6x + 2x 2 ||

What is a monomial? A **monomial** is a variable (such as x), a number (such as 17) or the product of a variable and a number (such as -3x 2 ) where the exponents are whole numbers.

So what is not a monomial? A term that has a variable with a negative exponent (such as x -2 ), a fractional term with a variable as a denominator (such as 3/x), or an exponent that is a variable (such as 7 y ).

What is a polynomial? A polynomial is a collection of terms (monomials in particular) that are being added or subtracted. Are all expressions polynomials? No way! There are some characteristics where a collection of terms would just be an expression, but could not be classified as a polynomial if an expression contains any of the following: a variable with a negative exponent, a fractional term with a variable as a denominator, or an exponent that is a variable.

Which of the following expressions can be called polynomials?

a) 5x 2 + 8x -3 b) 9x c) -x 6 + 9x - 8 d) 2x 3 + 5 x

Expressions b and c are polynomials. Expressions a and d cannot be called polynomials. Expression a has a variable with a negative exponent, and expression d has an exponent that is a variable.

**Polynomial Basics**

 * ** Classifying ** : A polynomial is a monomial if it consists of a single term, a binomial if it has two terms, and a trinomial if it has three terms
 * Monomial: 6//x// 2 ; Binomial: 2//x// 3 + 7; Trinomial: -5//x// 2 + 8//x// - 10
 * **Degree of a monomial (term)**: The exponent on the variable of the term or the sum of the exponents on the variables within a term.
 * 7x 2 - degree of 2; 4 - degree of 0; 2x 2 y 3 - degree of 5

When writing in descending order, make sure the exponents decrease in value from left to right. Since the term 7//x// 3 has a degree of 3, this term would be written first. Next would be -//x// 2, because it has a degree of 2, then we would write 12//x// after because this term has a degree of 1 (remember that the x has an invisible exponent of 1), and then finally -4 because it is a constant and has no variable (not sure about writing how -4 is the same as -4x 0 ???). So the polynomial written in descending order would look like this: 7//x// 3 - //x// 2 + 12//x// - 4.
 * **Descending Order:** Writing the terms of a polynomial in an order where the exponent (degree) for each term decrease from left to right
 * Write the following polynomial in descending order: 12//x// + 7//x// 3 - 4 - //x// 2


 * **Degree of a polynomial**: The value of the highest degree of the terms
 * Given the trinomial, 6//x// - 13//x// 4 + 12, the degree would be 4 since that is the value of the highest exponent for each of the three terms. Typically a polynomial is written in descending order, and if that is the case then the degree can be found from the first term.


 * **Leading Coefficient**: The coefficient of the first, or leading, term of a polynomial when it is written is descending order

First write the polynomial in descending order so that the degree and leading coefficient are easily identifiable: 3//x// 2 + 8//x// - 15. The first term provides the degree and the leading coefficient. Since the value of the exponent of the first term is 2, then the degree of the polynomial would be 2. The coefficient that "leads" the polynomial is 3, since that is the coefficient of the first term. Finally, since this polynomial has three terms, it would be classified as a trinomial.
 * Example:** Given the polynomial 8//x// + 3//x// 2 - 15, write in descending order, classify the polynomial, and give the degree and leading coefficient.

1) Classify the polynomial: 9//x// 2 - 1
 * Polynomial Basics Quick Check:**

2) Write the following polynomial in descending order: -5//x// - //x// 3 + 14 + 6//x// 2 . Then determine the degree and leading coefficient.

3) Create a binomial that has a degree of 4 and a leading coefficient of 3.


 * Polynomial Basics Quick Check Solutions**

**Adding and Subtracting Polynomials**
When adding and subtracting polynomials, combine the like terms. Remember that like terms need to have the same variable(s) and the same exponent. For example, you can combine 5//x// 2 and -3//x// 2 because each term has the variable "//x//" with a degree of 2 (exponent of 2). Can you add 7//m// 3 and 6//m// 4 ? Nope! Although both terms have the same variable, they do not have the same exponent, so you cannot add the two together. Sometimes it can be helpful to underline your like terms before you add/subtract them to help keep things organized! Watch the videos below to see examples of how to add and subtract polynomials. media type="youtube" key="L6EhaBEdjH0" width="560" height="315"

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 * Adding and Subtracting Polynomials Quick Check:**

1) (3//x// 2 + 5//x// - 10) + (8//x// 2 - 3//x// + 2)

2) (5//x// 2 - 4//x// + 6) - (4//x// 2 + 4//x// + 6)


 * Adding and Subtracting Polynomials Quick Check Solutions**


 * Multiplying Polynomials **
 * ** (Monomial)(Polynomial): ** Multiply the monomial on the outside of the parentheses by every term on the inside of the parentheses. The number of terms of your final answer will match the number of terms of the polynomial that is on the inside of the parentheses.
 * ** (Binomial)(Binomial): ** Multiply each term of the first binomial by each term of the second binomial (double distribution). Since there are two terms for each binomial, you will use multiplication 4 times. Make sure to combine any like terms after you have multiplied the binomials.



**Multiplying Polynomials Quick Check:** Multiply the following. 1) -6x(2x 2 + 3x - 10) 2) (5x + 7)(4x - 9) 3) (3x - 9)(x 2 - 2x + 15) **Multiplying Polynomials Quick Check Solutions**
 * ** (Binomial)(Trinomial): ** Multiply each term of the binomial to each term of the trinomial. Since a binomial has 2 terms and a trinomial has 3 terms, you will use multiplication 6 times. Make sure to combine like terms after you have multiplied the polynomials. Check out the example below.

Applications of Polynomial Expressions
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 * Applications of Polynomials Quick Check:**

A floor length mirror has dimensions 2' by 5' surrounded by a uniform frame of an unknown width. Write expressions to represent the perimeter of the frame and the combined area of the mirror and frame.


 * Applications of Polynomials Quick Check Solutions**

Polynomial PEMDAS
What happens when we need to perform multiple operations on polynomials? What order do we follow? That's right! We still follow the order of operations when performing operations on polynomials. Watch the videos below for a few examples on how to apply PEMDAS to polynomials.

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1) (3x + 2)(2x - 9) - 4(x 2 + 6x - 1)
 * Polynomial PEMDAS Quick Check:**


 * Polynomial PEMDAS Quick Check Solutions**

**Using Function Notation**
Sometimes polynomials can represent different types functions (this be learned in further detail in the Quadratics units). We can use function notation to evaluate different polynomials, and function notation can be used when combining different functions together. Recall that function notation looks like so: f(x) = 3x - 2, which is a linear function, but now we are including functions with a degree higher than 1. Take the functions j(x) = 9x + 1 and k(x) = -x 2 - 5x + 6. To find j(x) + k(x), just substitute the polynomials in for the functions, and add. j(x) + k(x) (9x + 1) + (-x 2 - 5x + 6) -x 2 + 4x + 7 so j(x) + k(x) = -x 2 + 4x + 7 Do you recall how to evaluate a function? If asked to find k(3), for example, substitute in 3 for each x in the function k(x) and evaluate. Find k(3) for k(x) = -x 2 - 5x + 6 k(3) = -(3) 2 - 5(3) + 6 k(3) = -9 - 15 + 6 so k(3) = 18 Use the following functions to complete the quick check: math f(x) = 3x - 4; g(x) = 8x + 1; h(x) = 2x^{2} - 5x + 3; j(x) = -x^{2} + 9x - 1\ math
 * Using Function Notation Quick Check:**

math 1) j(x)-h(x)\ math math 2) g(x)\cdot h(x)\ math math 3) f(x) \cdot g(x) -j(x)\ math


 * Using Function Notation Quick Check Solutions**