Introduction to Polynomials

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Welcome to Algebra I. Today I will introduce you some basic ideas about polynomial. You will be able to recognize polynomials and do some basic calculations about polynomials after our lesson.

Now, Let’s

Start

Our lesson consists of 4 parts:

Introduction to polynomial

Addition & subtraction of polynomials

Multiplication & division of monomials

QUIZ

1. 3.

2. 4.

Proceed in order

QUIZ 3

QUIZ 2

QUIZ 1

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What is a POLYNOMIAL Exactly ?

First,

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Definition: In mathematics, a polynomial is an expression

consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

Here is the answer:

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Um… it seems quite abstract and useless, isn’t it?

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So, Why do we need to learn polynomial?

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Well, Let’s watch a short video

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https://www.youtube.com/watch?v=1LcSTV6n0Ac

No matter you want to be in

engineering, business,

science, education etc. in

the future, polynomial will

be useful for you.

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Now let’s come back to polynomial

The definition might be a little bit confusing so let’s take a look at a couple of examples.

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2x²+1 5y³+3y²-5 4z²+z x³y²+2x²y-5x+6

Examples

Can you relate each part of the examples to the definition of polynomial?

Now just think:

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Definition: In mathematics, a polynomial is an expression

consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

x³y²+2x²y-5x+6

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There is a special kind of polynomial called monomial which only has one term and does not involve any addition and subtraction.

Example:• 3x²y• 2x³• 5x²yz³

Proceed to Quiz 1

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QUIZ Time

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(2 Questions)

Which one of the following is a polynomial?

Question 1

A. 2/3x3 - 3x2 + 5y

B. 2x½ + x2y

C. x3 - 2x-2 + y

D. 2x3 - 2x/y

GOOD JOB

Next question

WRONG

Back to the question

Which one of the following is NOT a polynomial?

Question 2

A. x2 + xy + 5y

B. 2x3 - 3y

C. 2x3 - 2x2 + 3xy-²

D. x³ + xy + 1/3y

GOOD JOB

Proceed to the next

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WRONG

Back to the question

Now you just had a brief understanding of what a polynomial exactly is. So let’s keep on moving!

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What we already know?

• Addition Example: 5+6=11

• Subtraction Example: 34-12=22

Addition & Subtraction of polynomials

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To add polynomials we simply add any like terms together

Like terms are terms whose variables (and their exponents) are the same.

Note: the coefficients can be different

Example: 3x³y² and 5x³y² are like terms

How do we approach this to polynomials?

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When we add the like terms, we add the coefficients of the like terms and leave the variables (and their exponents) unchanged.

Example: • 3x³y² + 5x³y² = (3+5)x³y² = 8x³y²

• (3x³y²+2xy²+5) + (3x²y+4xy²+3)

=3x³y²+3x²y+(2+4)xy²+(5+3)

=3x³y²+3x²y+6xy²+8

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The strategy for subtraction is similar

Example:

• (3x³y²+2xy²+5) - (3x²y+4xy²+3)

=3x³y²+3x²y+(2-4)xy²+(5-3)

=3x³y²+3x²y-2xy²+2

Proceed to quiz 2

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QUIZ Time

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(3 Questions)

Add the polynomials (3x2 - 6x + xy), (2x3 - 5x2 -3y) and (7x + 8y)

Question 1

A. 2x3 - 8x2 + x + xy + 5y

B. 2x3 - 2x2 + x + xy - 5y

C. 2x3 - 2x2 - x + xy + 5y

D. 2x3 - 2x2 + x + xy + 5y

GOOD JOB

Next question

WRONG

Back to the question

Subtract (-3x2 + 5y - 4xy + y²) from (2x2 - 4y + 7xy - 6y²)

Question 2

A. -5x2 + 9y - 11xy + 7y2

B. 5x2 - 9y + 11xy - 7y2

C. -x2 + y + 3xy - 5y2

D. -x2 - 9y + 11xy - 7y2

GOOD JOB

Next question

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Back to the question

If P = 5x4 - 2x2  + 4x - 3 and Q = 5x4 + 3x3 - 4x + 3, what is P - Q?

Question 3

A. -3x3 - 5x2  + 8x  - 6

B. -3x3 - 2x2  + 8x  - 6

C. -3x3 + 2x2  + 8x  - 6

D. -3x3 - 2x2

GOOD JOB

Proceed to the next

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WRONG

Back to the question

What we already know?

• Multiplication Example: 2x4=2¹x2²=2³=8

• Division Example: 27÷3=3³÷3¹=3²=9

Multiplication & Division of monomials

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To multiply monomials we multiply the coefficients of the monomials and add the exponents of each unknown of the variables.

Example:• 2y·x³y = 2y¹·x³y¹ = 2x³y²• 2xy²·3x²y = 2x¹y²·3x²y¹ = 6x³y³

Note: The result of multiplication of monomials must be a monomial.

Multiplication

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The strategy for Division is similar to multiplication. We divide the coefficients of the monomials and subtract the exponents of each unknown of the variables

Example:• 2x³y÷y = 2x³y¹÷y¹ = x³yº = x³• 2x³y²÷3x²y = 2x³y²÷3x²y¹ = 2/3xy

Note: The result of division of monomials does NOT have to be a polynomial.

Division

Proceed to quiz 3

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QUIZ Time

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(3 Questions)

Multiply (2x²y) with (3xy³)

Question 1

A. 5x²y³

B. 5x3y4

C. 6x3y4

D. 2x³+3y4

 

GOOD JOB

Next question

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Back to the question

Divide (5x²y4) by (2xy)

Question 2

A. 3xy³

B. 5/2xy³

C. 10x3y5

D. 5/2x²y4

 

GOOD JOB

Next question

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Back to the question

Divide (3x²y4) by (5x³y³)

Question 3

A. 3/5x-1y4/3

B. 5/3xy

C. 3/5xy

D. 3/5x-1y

 

GOOD JOB

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move on

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Back to the question

Congratulations! You have already finished the lesson. Now you can go back to review the lesson and retake the quiz if you didn’t do well.

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Quiz