Unit 1: Polynomials

Unit 1: Polynomials

Classifying, Adding, Subtracting, and Multiplying

• One term: Monomial

• Two terms: Binomial

• Three terms: Trinomial

• Four or more terms: Polynomial

By number of terms:
(Terms are separated by a + or a - symbol)

• Constant: highest exponent is 0

  • (looks like a plain number)

• Linear: highest exponent is 1

  • (looks like a plain variable)

• Quadratic: highest exponent is 2

• Cubic: highest exponent is 3

• Quartic: highest exponent is 4

By degree (highest exponent)

Classifying Polynomials

Polynomials can be classified two ways:

Drag and Drop

The polynomial $$5-3x+2x^2$$ would be classified by degree as​

Placeholder

because its highest exponent is​

Placeholder

.

Drag these tiles and drop them in the correct blank above

quadratic

2

quartic

4

constant

0

3

cubic

trinomial

monomial

Match

Match the following polynomials to their classifications:

• x² - 9

• 2x²

• 13x

• 3 - 2x

• 5x - 3x² + 11

Quadratic Binomial

Quadratic Monomial

Linear Monomial

Linear Binomial

Quadratic Trinomial

• Quadratic Binomial

• Quadratic Monomial

• Linear Monomial

• Linear Binomial

• Quadratic Trinomial

Dropdown

16a²b⁵

How many terms does this polynomial have?

​ ​

Select answer

Drag and Drop

The polynomial $$7x-6$$ would be classified by degree as ​​

Placeholder

because its highest exponent is ​

Placeholder

.

Drag these tiles and drop them in the correct blank above

linear

1

quadratic

4

constant

3

cubic

binomial

2

0

Drag and Drop

The expression $$2x^2+8x^3-7x+1$$ would be classified by number of terms as a ​

Placeholder

because it has

Placeholder

terms.

Drag these tiles and drop them in the correct blank above

polynomial

4 or more

trinomial

3

binomial

2

monomial

1

cubic

quadratic

Drag and Drop

3x⁴ - 2x⁵ + 7x² - 12



The degree of the polynomial is ​

Placeholder

Drag these tiles and drop them in the correct blank above

5

3

4

-2

7

2

-12

Multiple Choice

What is the degree of the following polynomial:

-6

• 0

• 1

• -6

Writing a Polynomial in Standard Form

The terms are written from "highest" exponent to "lowest exponent.
The constant should always be written last.

Example: The polynomial -5 + 2x³ + 4x - 6x² would be re-written in standard form as:
2x³ - 6x² + 4x -5

When written this way, the exponents in each term go downwards in the order 3, 2, 1 (unseen), 0 (unseen)

Reorder

Click and drag to rearrange the terms into a polynomial written in STANDARD FORM:

• $$-3x^3$$

• $$-4x^2$$

• $$+9x$$

• $$-5$$

1

2

3

4

• 1
• 2
• 3
• 4

Reorder

Click and drag to rearrange the terms into a polynomial written in STANDARD FORM:

• $$-6x^5$$

• $$+2x^4$$

• $$-7x^2$$

• $$+3x$$

• $$-12$$

1

2

3

4

5

• 1
• 2
• 3
• 4
• 5

Adding Polynomials

Combine Like Terms!

"Like terms" have the same variable AND the same exponent.

​Like terms:

NOT Like terms:

• 3x and 2x

• 4x² and -6x²

• -y⁴ and -6y⁴

• etc.

• 5x and 6y (different variables)

• 2x² and -9x (different exponents)

• etc.

Adding Polynomials

1. Combine x² terms: -2x² + 5x² = 3x²

2. Combine x terms: 3x - 2x = x

3. Combine constants: -4 + 8 = 4

Write your answer in standard form: 3x² + x + 4

Math Response

Add the following polynomials by combining like terms:

$$\left(-2x^3+3x^2-4x-6\right)+\left(4x^3-7x^2+6x-8\right)$$

Type the sum into the answer box

Type answer here

Correct Answer: 2x^3-4x^2+2x-14

Deg^°

Rad

Math Response

Add the following polynomials by combining like terms:

$$\left(5x^3+3x^2-x+10\right)+\left(8x^2+6x-2\right)$$

Type the sum into the answer box

Type answer here

Correct Answer: 5x^3+11x^2+5x+8

Deg^°

Rad

Math Response

Add the following polynomials by combining like terms:

$$\left(6x^3+12x^2-5x-8\right)+\left(2x^2-4x^3+9x-12\right)$$

Type the sum into the answer box

Type answer here

Correct Answer: 2x^3+14x^2+4x-20

Deg^°

Rad

Subtracting Polynomials

1. KEEP the first polynomial exactly the way it is.

2. SWITCH the sign between the polynomials from a - to a +.

3. CHANGE the sign for ALL of the terms in the second polynomial. Positive terms become negative, negative terms become positive.

4. COMBINE like terms.

Subtracting Polynomials

​Original Problem:

​Rewrite:

​Keep

Switch to +

Change all signs

Combine like terms:

Math Response

Subtract the following polynomials:

$$\left(2x^2-4x-6\right)-\left(-4x+9+7x^2\right)$$

Type answer here

Correct Answer: -5x^2-15

Deg^°

Rad

Math Response

Subtract the following polynomials:

$$\left(-4x^2+3x+1\right)-\left(-7x^2+4x-9\right)$$

Type answer here

Correct Answer: 3x^2-x+10

Deg^°

Rad

Math Response

Subtract the following polynomials:

$$\left(3x^2+5x-7\right)-\left(x^2-3x+4\right)$$

Type answer here

Correct Answer: 2x^2+8x-11

Deg^°

Rad

Multiplying Polynomials: Distribution

​Whenever you multiply terms:

MULTIPLY the coefficients (number in front of a variable) and ADD the exponents.

Math Response

Use Distributive Property to solve.

x³(2x²+3x)

Type answer here

Correct Answer: 2x^5+3x^4

Deg^°

Rad

Math Response

Multiply 4x(3x⁷ + 2x⁵)

Type answer here

Correct Answer: 12x^8+8x^6

Deg^°

Rad

Math Response

-5x⁶(3x² - 12x + 30)

Type answer here

Correct Answer: -15x^8+60x^7-150x^6

Deg^°

Rad

Multiplying Polynomials: FOIL Method

​The FOIL method is used to multiply two binomials together.

​1. Multiply the first terms from each binomial
2. Multiply the "outside" terms from each binomial
3. Multiply the "inside" terms from each binomial
4. Multiply the last terms from each binomial

Add all the products from steps 1-4
Combine like terms
Write your answer in standard form

Multiplying Polynomials: FOIL Method

​ Remember that squaring something means to multiply it by itself.

Therefore, squaring a binomial requires the FOIL Method.

​means

Math Response

(x + 7)(x + 9)

Type answer here

Correct Answer: x^2+16x+63

Deg^°

Rad

Math Response

(x - 3)(x +10)

Type answer here

Correct Answer: x^2+7x-30

Deg^°

Rad

Math Response

(x - 7)(x + 7)

Type answer here

Correct Answer: x^2-49

Deg^°

Rad

Math Response

Expand the squared binomial
(x - 5)²

Type answer here

Correct Answer: x^2-10x+25

Deg^°

Rad

Multiplying Polynomials: Binomial * Trinomial

​Distribute the first term of the binomial to each term in the trinomial

Distribute the second term of the binomial to each term in the trinomial

Combine like terms

Write your answer in standard form

Math Response

Multiply:

$$\left(3x-2\right)\left(5x^2+4x-1\right)$$

Type answer here

Correct Answer: 15x^3+2x^2-9x+2

Deg^°

Rad

Math Response

$$\left(x+2\right)\left(2x^2+4x+9\right)$$

Type answer here

Correct Answer: 2x^3+8x^2+13x+18

Deg^°

Rad