Lesson Objectives

- Learn how to determine if two or more terms are "like terms"
- Learn the basic definition of a polynomial
- Learn how to find the degree of a polynomial
- Learn how to write a polynomial in standard form

## What is a Polynomial?

Before we get into the definition of a polynomial, it is important to understand the definition of a "term". A term is a number, variable, or the product of a number and one or more variables raised to powers. Terms are separated by "+" or "-" signs. In our above picture, we have the algebraic expression:

7x

The 7x

"Like Terms" occur when two or more terms have exactly the same variable parts. This means the variable(s) is(are) the same and raised to the same power(s).

5x

9x

2x

11x

We can add or subtract "like terms" by keeping the variable part the same and performing operations with the coefficients. We are just using the distributive property.

3x - 5x = (3 - 5)x = -2x

13x

Examples of Polynomials: $$11$$ $$-2x^3-5x$$ $$12x^2-7x+1$$ Examples of Non-Polynomials: $$\frac{2x - 5}{x^2}$$ $$5x^{-4}+ 1$$ $$\sqrt{7 - x}$$ Some polynomials occur so often, that we give them special names. A monomial is a polynomial with only one term.

Examples of Monomials:

8x

12x

-6x

A binomial is a polynomial with two terms.

Examples of Binomials:

9x

19x

-11x

Examples of Trinomials:

25x

-5x

3x

9x + 7x

We can rewrite this polynomial in standard form:

7x

Notice how the first or leftmost term "7x

12x » The degree is 1

4x

-10x

7x

4 » The degree is 0 (4x

The largest degree of any non-zero term in the polynomial is the degree of the polynomial.

7x

x

x

2x

3x

-5x

10 » degree of 0

So it's really straight forward that 3x

7x

^{2}+ 6x - 9The 7x

^{2}, 6x, and 9 are terms. Notice how they are separated by "+" or "-" signs."Like Terms" occur when two or more terms have exactly the same variable parts. This means the variable(s) is(are) the same and raised to the same power(s).

5x

^{2}, 9x^{2}» Like Terms: each has the variable x raised to the 2nd power.9x

^{3}y^{2}, 15x^{3}y^{2}» Like Terms: each has the variable x raised to the 3rd power and the variable y raised to the 2nd power.2x

^{4}, 13x » Not Like Terms: each has the variable x, but our exponents are not the same.11x

^{3}, 14y^{3}» Not Like Terms: although the exponents are the same, the variables are different.We can add or subtract "like terms" by keeping the variable part the same and performing operations with the coefficients. We are just using the distributive property.

3x - 5x = (3 - 5)x = -2x

13x

^{2}+ 9x^{2}= (13 + 9)x^{2}= 22x^{2}### What is a Polynomial

A polynomial is either a single term or a finite sum of terms where all variables have whole number exponents and there are no variables in any denominator. A polynomial is the simplest type of algebraic expression.Examples of Polynomials: $$11$$ $$-2x^3-5x$$ $$12x^2-7x+1$$ Examples of Non-Polynomials: $$\frac{2x - 5}{x^2}$$ $$5x^{-4}+ 1$$ $$\sqrt{7 - x}$$ Some polynomials occur so often, that we give them special names. A monomial is a polynomial with only one term.

Examples of Monomials:

8x

^{7}12x

^{4}-6x

^{5}y^{3}A binomial is a polynomial with two terms.

Examples of Binomials:

9x

^{4}- 219x

^{3}- 4x^{2}-11x

^{5}+ 7Examples of Trinomials:

25x

^{3}+ 2x - 14-5x

^{2}- 12x + 13x

^{9}+ 4x^{5}+ 6### Standard Form of a Polynomial

A polynomial is written in standard form when the powers are in descending order.9x + 7x

^{4}- 2 + 12x^{2}- 3x^{3}We can rewrite this polynomial in standard form:

7x

^{4}- 3x^{3}+ 12x^{2}+ 9x - 2Notice how the first or leftmost term "7x

^{4}" has the largest exponent on x. Then each position that follows has the next largest exponent on x. The final term (rightmost) is a 2. We can think about this as 2x^{0}since x^{0}is 1 and 2 • 1 is 2. Generally speaking, we are expected to write any polynomial answer in standard form.### The Degree of a Polynomial

The degree of a term in a polynomial is the sum of the exponents on the variable(s) of the term.12x » The degree is 1

4x

^{9}» The degree is 9-10x

^{5}y^{3}» The degree is 87x

^{3}y^{7}z^{14}» The degree is 244 » The degree is 0 (4x

^{0}= 4 • 1 = 4)The largest degree of any non-zero term in the polynomial is the degree of the polynomial.

7x

^{2}- x + 1 » degree of 2x

^{3}y^{2}+ 4xy - 3 » degree of 5x

^{18}y^{3}z^{5}- 9x^{2}y^{3}z + 14 » degree of 26### Standard Form of a Polynomial with More Than One Variable

What happens when we want to write a polynomial with two variables in standard form? For example: $$2x^4y+3x^3y^3-5x^2y^3+10$$ In this case, we still list the term with the highest degree first, then the next highest, so on and so forth.2x

^{4}y » degree of 53x

^{3}y^{3}» degree of 6-5x

^{2}y^{3}» degree of 510 » degree of 0

So it's really straight forward that 3x

^{3}y^{3}goes first, but what happens here when we have a tie? This won't happen with one variable, because in that case, we would have like terms and be able to simplify. So here what we do is think about alphabetical order. Since x comes before y, we would think about which term has the x with the larger exponent. 2x^{4}y has a larger exponent on x, so it will be listed first. $$3x^3y^3+2x^4y-5x^2y^3+10$$#### Skills Check:

Example #1

Determine which pair of terms are "not like terms"

Please choose the best answer.

A

$$3x^2, 5x^2$$

B

$$7xyz, -2zxy$$

C

$$5x^2y^2z, -4zx^2y^2$$

D

$$6x^2yz, 5y^2xz$$

E

$$-x^2y^4, 15y^4x^2$$

Example #2

Determine which is "not a polynomial"

Please choose the best answer.

A

$$5$$

B

$$4x^{-2}- 8x - 1$$

C

$$-x^2 + x - 7$$

D

$$\frac{3x^2 - 5}{2}$$

E

$$\sqrt{3}\cdot x^3 - 8$$

Example #3

Find the degree of the polynomial $$5x^2y^9 + 7xy^{15}- 12x^{19}$$

Please choose the best answer.

A

$$11$$

B

$$16$$

C

$$38$$

D

$$1$$

E

$$19$$

Example #4

Write in standard form $$3x^5 + 9x^9 - 4 + 11x$$

Please choose the best answer.

A

$$3x^5 + 9x^9 - 4 + 11x$$

B

$$11x + 3x^5 + 9x^9 + 4$$

C

$$9x^9 - 4 + 11x + 3x^5$$

D

$$9x^9 + 3x^5 + 11x - 4$$

E

$$9x^9 + 11x - 3x^5 - 4$$

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