Chapter 4.1

Objectives

Review the lesson material
Defeat the Area challenge
Complete the lesson reflection
Submit the assignment

Key Vocabulary

Polynomial

A mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.
Rational Expression

An expression that can be written as the ratio of two polynomials.
Closed Under Operation

A property of a set where performing an operation on members of the set results in an outcome that is also a member of the set.
Degree of a Polynomial

The highest power of the variable in the polynomial.

Lesson Outcomes & Standards

Student will correctly add, subtract, and multiply given polynomials and demonstrate their understanding by solving at least 5 problems with 80% accuracy.
Students will explain, in their own words, how polynomials are closed under addition, subtraction, and multiplication and provide at least two examples for each operation

Common Core:

A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

The story continues...

Welcome to Polynomial Park, the theme park of Quantitown and what many consider to be the most fun place in town… at least it used to be. The theme park is barely running with all of the sabotage the Fraction Faction have done. Rides aren’t running, attractions are ruined, and the Ringmaster is telling me the circus won’t be able to start.

I need you to figure out what is going on and get it fixed. The show must go on!

Taking a twist for a loop

rollercoaster — **Figure 1.** *Theme park ride designers are experts in physics and polynomials.*

"The Beast" is the world's largest wooden rollercoaster in Kings Island, Ohio. Despite being made of wood, it sprawls over 35 acres of wooded terrain at speeds up to 65 miles per hour. Every so often, they extend this rollercoaster to maintain its place as a world record holder. The architects and engineers of this marvel of carpentry made extensive use of polynomials to ensure that the heart-stopping drops and curves would be exhilarating but also safe.

By performing operations like addition, subtraction, multiplication, and division on polynomials, rollercoaster designers are able to design incredibly interesting rides as they bring their graphs into 3-dimensional space.

The Essential Nature of Polynomials

Polynomials are a lot like the numbers we use every day. Just like how we can add, subtract, and multiply numbers, we can do the same with polynomials. Something special happens when we work with polynomials in these ways: we always end up with another polynomial. This is because polynomials are closed under addition, subtraction, and multiplication. This means that no matter how we mix or change them using these operations, they still remain as polynomials. It's important to remember this basic rule because it helps us predict and understand what happens when we work with polynomials, just like we understand what happens when we add or multiply regular numbers.

Closed under the operations?

If a set is closed under an operation, it means that whenever you apply that operation to any elements within the set, the result is also an element of that set.

For example, consider the set of even numbers. If you add any two even numbers together (like 4 + 6 = 10), the result is always another even number. Therefore, we can say that the set of even numbers is closed under addition. This would not be the case if you added two odd numbers together (like 3 + 5 = 8). When you add two odd numbers, you get an even number, so the sum of two odd numbers is not closed under addition.

Polynomials stay polynomials for addition, subtraction, and multiplation.

Let's explore the concept that polynomials are closed under the operations of addition, subtraction, and multiplication.

Addition: If you add any two polynomials, the sum is always a polynomial:

- and add together to give us $LaTeX: 3x^2+2x\:+5$ , which is a polynomial.

Subtraction: If you subtract one polynomial from another, the result is always another polynomial.

- Subtracting from add together to give us $LaTeX: 2x^2+6x\:-3$ , which is a polynomial.

Multiplying: Multiplying any two polynomials results in another polynomial.

- $LaTeX: \left(x+1\right)\times\left(x^2-x+3\right)$ gives us , which simplifies to , which is a polynomial.

Polynomials are not closed under division.

Dividing: Polynomials are not closed under division! Sometimes, they result in a rational expression instead. In other words, it could have a remainder.

- Dividing the polynomial by gives us $LaTeX: x+\frac{1}{x-1}$

Polynomials do not involve division by variables, and they don't have any undefined points. In the example, if x=1, then you would be dividing by zero, which is undefined. Polynomials are never undefined.

Polynomials vs Rational Expressions

While we are on the topic of closed expressions, let's explore why rational expressions are not polynomials.

Comparing Polynomials and Rational Expressions
Feature	Polynomial	Rational Expression
Form	$LaTeX: P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$	, where $LaTeX: P\left(x\right)$ and $LaTeX: Q\left(x\right)$ are polynomials
Division by Variables	No	Yes
Examples		$LaTeX: \frac{3x^2 + 2x + 1}{x - 1}$
Domain	All real numbers	All real numbers except where the denominator is zero
Closed under Operations	Addition, subtraction, and multiplication	Addition, subtraction, multiplication, and division
Undefined Points	None	Points where the denominator is zero
Structure	Sum of terms with non-negative integer exponents	Ratio of two polynomials

The Polynomial of Degree 0

A polynomial of degree zero is when you end up with a coefficient of zero in front of all variables. For example, LaTeX: 0x^2+0x+7 would simplify to be 7, but it would still be a polynomial. It is simply a polynomial of degree zero.

Master Essential Skills

Question 1: Multiply the following polynomials. Is the result a polynomial? If so, what degree?

$LaTeX: \left(3x^2+2\right)\times\left(x-7\right)$

Question 2: Add the following. Is the result a polynomial?

$LaTeX: \left(3x^2-x\:+5\right)+\left(-3x^2+x-5\right)$

Check Your Work

Answer A: (Hint: use foiling)

LaTeX: 3x^3-21x^2+2x-14

Foiling 3x squared term being multiplied to both the x and the -7, while the 2 is being multiplied by the x and -7

Yes, this is a 3rd-degree polynomial.

Answer B:

Since these are equal but opposite, each term cancels out.

Therefore, the answer is 0.

This is a polynomial of degree zero.

Optional: Dividing Polynomials (Preview)

Video Activity: 15 minutes

Dividing polynomials with long division takes several steps, but this video will show you how to do it with a familiar process. This topic will be covered later, but if provided to give you a head start. Watch it and follow along so you will be better prepared when it's your turn to try!

Knowledge Check

Answer the following questions.

How does dividing polynomials relate to dividing integers?

Check Answers

Answer A:

Dividing polynomials and integers have similar outcomes—both yield a quotient and a remainder.

The process also involves dividing the leading terms, multiplying, and subtracting step-by-step. Even though polynomial division is a little more complicated due to the variables and their degrees, their operations mirror those of "normal" numbers like integers.

Practice

Add polynomials

The length of a rectangle is given by x²+4x+2, and the width is given by 4x. Express the perimeter of the rectangle as a trinomial.

Solution:

The perimeter (P) of a rectangle is twice the length (l) added to twice the width (w).

P=2L+2W. Diagram showing that there are two widths and two lengths that build the formula.

Step 1

Write the given length and width as:

l=x²+4x+2

w=4x

Step 2

Plug in the given expressions into the formula.

P=2l+2w

$LaTeX: P=2\left(x^2+4x+2\right)+2\left(4x\right)$

Step 3

Use distributive property to expand the terms and remove any brackets.

i.e., multiply 2 to each term within the brackets.

LaTeX: P=2x^2+8x+4+8x

Step 4

Combine the like terms together.

8x and 8x are like terms, and their coefficients can be added together.

LaTeX: P=2x^2+8x+8x+4

LaTeX: P=2x^2+16x+4

2x² and 4 do not have any other like terms in this expression.

Step 5

Check if this expression is a trinomial.

It has 3 terms (2x², 16x, and 4) and hence is a trinomial.

Conclusion

The perimeter of the given rectangle is a trinomial of the form 2x²+16x+4.

Simplify by adding/ subtracting polynomials

Simplify the following expression:

$LaTeX: 3z^3+9z^4+9z-\left(2z^2+7z^3+4z^2\right)$

Solution:

The given expression is a polynomial.

Once simplified, a single variable should have only one term with the variable raised to each power.

Step 1

Use the distributive property to expand the terms and remove any brackets.

i.e., multiply -1 to each term within the bracket.

LaTeX: 3z^3+9z^4+9z-2z^2-7z^3-4z^2

Step 2

Combine the like terms together.

3z³ and -7z³ are like terms.

-2z² and -4z² are like terms.

LaTeX: 3z^3-7z^3+9z^4+9z-2z^2-4z^2

Step 3

Add the coefficients of the like terms.

$LaTeX: \left(3-7\right)z^3+9z^4+9z-\left(2+4\right)z^2$

LaTeX: -4z^3+9z^4+9z-6z^2

Step 4

Sort the terms by the exponent on z, in descending order.

LaTeX: 9z^4-4z^3-6z^2+9z

Step 5

Check if the variable z has only one term with the variable raised to each power.

Conclusion

The simplified expression is

LaTeX: 9z^4-4z^3-6z^2+9z

Multiply polynomials

A triangular garden bed has it's base as 3x+2 and height as 5xy²+xy+3. Calculate the area of the triangular garden bed.

Solution:

The area (A) of a triangle is half the product of it's base (b) and height (h).

Diagram showing the base and height of a triangle.

$LaTeX: A=\frac{1}{2}bh$

Step 1

Write the given base and height as:

LaTeX: b=3x+2

LaTeX: h=5xy^2+xy+3

Step 2

Plug in the given expressions into the area formula:

$LaTeX: A=\frac{1}{2}bh$

$LaTeX: A=\frac{1}{2}\left(3x+2\right)\left(5xy^2+xy+3\right)$

Step 3

Use distributive property to multiply the polynomials and thereby expand the terms and remove any brackets.

Distributive property

Remember to:

-Multiply the coefficients together.

-Use the product rule to multiply similar variables, i.e., add the exponents of similar bases. (a^m•aⁿ=a^m+n)

Multiply 3x to each term within the second set of brackets (5xy²+xy+3), to obtain new terms.

$LaTeX: 3x\times5xy^2=\left(3\times5\right)\cdot\left(x\times x\right)\cdot y^2=15x^2y^2$

$LaTeX: 3x\times xy=3\cdot\left(x\times x\right)\cdot y=3x^2y$

$LaTeX: 3x\times3=\left(3\times3\right)\cdot x=9x$

Multiply 2 to each term within the second set of brackets (5xy²+xy+3), to obtain new terms.

$LaTeX: 2\times5xy^2=\left(2\times5\right)xy^2=10xy^2$

$LaTeX: 2\times xy=2xy$

$LaTeX: 2\times3=6$

Step 4

Add all the new terms together, and rewrite the area formula:

$LaTeX: A=\frac{1}{2}\left(15x^2y^2+3x^2y+9x+10xy^2+2xy+6\right)$

Step 5

Combine the like terms together:

$LaTeX: A=\frac{1}{2}\left(15x^2y^2+3x^2y+9x+10xy^2+2xy+6\right)$

There are no like terms.

Step 6

Sort the terms by the exponent on x and y in descending order.

$LaTeX: A=\frac{1}{2}\left(15x^2y^2+3x^2y+10xy^2+2xy+9x+6\right)$

Check if the variables x and y have only one term with the variables raised to each power.

Step 7

Divide each term by 2:

$LaTeX: A=\frac{15}{2}x^2y^2+\frac{3}{2}x^2y+5xy^2+xy+\frac{9}{2}x+3$

Conclusion

The area of the triangular garden bed is

$LaTeX: A=\frac{15}{2}x^2y^2+\frac{3}{2}x^2y+5xy^2+xy+\frac{9}{2}x+3$

Gridiron Greats: Breaking Down Stats to Find Football’s MVP

rollercoaster — **Figure 2.** Each player has stats that determine their career. What are your stats?

There are 24 positions in football, and they all have a different job. The most recognizable positions are the scoring positions - quarterback, wide receiver, and running back. These are the players most likely to score points.

With so many positions, how can we determine the best player on the team? Player statistics help keep track of what a player does throughout the game - rushing yards, passing yards, touchdowns, tackles, sacks, etc. These stats can be combined at the end of each game, at the end of each season, and at the end of each player’s career to determine that player's contributions to the team.

We have to keep in mind that only like terms can be combined. It wouldn’t do us any good to add rushing yards and sacks or touchdowns and tackles.
.

Reflection Questions

When might you wish to use the quadratic formula versus another strategy (such as completing the square or factoring)?
What strategies might you use to check your work when using the quadratic formula?
Can you think of any real-world situations where the quadratic formula might be used?
How do you decide which of the two solutions to use in a given situation? Are there situations when you may need both?
Write in your own words how polynomials are closed under addition, subtraction, and multiplication.

What's next?

Complete the challenge area

Continue to the practice section.

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