Adding and Subtracting Polynomials

Key Concepts

• Understand polynomials.
• Add and Subtract Monomials
• Add and Subtract Polynomials
• Understand the application of polynomials to real life problems

Adding and Subtracting Polynomials

Monomial  

A monomial is a real number, variable, or the product of a real number and one or more variables with whole number exponents.  

The degree of a monomial is the sum of the exponents of the variables of a monomial.  

Why does a constant have a degree of 0? 

A constant has no variable. However, any number to the 0 power is 1, so you can multiply a constant by x^0 without changing the value of the constant. 

So, the degree of a constant is 0. 

Polynomial 

A polynomial is a monomial or the sum or difference of two or more monomials, called terms.  

Polynomials are named according to their degree. The degree of a polynomial is the greatest degree of any term of the polynomial. 

Polynomials are also named according to how many terms they have. 

Q: Why is 5x^3 – 4 is called a cubic polynomial? 

A: Because it has a degree of 3 and two terms 

Adding and Subtracting: Standard form of Polynomials 

What is the standard form of the polynomial? 

7x−5−x³+6x⁴−3x²

The standard form of a polynomial is the form of polynomial in which the terms are written in descending order according to their degree.        

Rewrite the polynomial in the standard form.            

The standard form of the polynomial is 

6x⁴−x³−3x²+7x−5

How to add and subtract Polynomials? 

Add and Subtract Monomials  

How can you use the properties of operations to combine like terms and write the expression in standard form? 

4x+ 3x²+2x+x²+5

1. Use algebra tiles to model the expression.  

2. Rearrange the tiles to group like terms together. 

The expression 4x+3x²+2x+x²+5 written in standard form is 4x²+6x+5

You can also rewrite the expression in standard form using the properties of operations. 

Add Polynomials 

How is adding polynomials like adding whole numbers? 

Consider the expressions 123 + 405 and  (x²+2x+3)+(4x²+5)

Before you add polynomials, the terms must be aligned with like terms. This is similar to how, before adding whole numbers, the numbers must be aligned according to their place value. 

Let’s solve some examples!  

Add Polynomials 

Example 1:  

What is the sum of (4x²+2x−3) and (3x²+6)

Solution:  

To add two polynomials, combine like terms.  

The sum of  (4x²+2x−3) and (3x²+6) is 7x²+2x+3

The sum of these two polynomials is a polynomial. 

Subtract Polynomials 

Example 2:  

What is the difference?  

(6x²+3x−2) – (3x²+5x−8)

Solution: 

To subtract two polynomials, subtract like terms.  

Method 1: Subtract vertically by lining up like terms. 

The difference of (6x²+3x−2) and (3x²+5x−8) and 3x²+5x−8 is 3x²−2x+6.

The difference of these two polynomials is also a polynomial.   

Apply Polynomials 

Example 3:  

An engineer is reviewing the layout of a solar farm. The solar farm shown has 4 small panels, 33 medium panels, and 32 large panels. What is the total area of the farm’s solar panels? 

Solution: 

Write an expression to represent the total area of the solar panels. 

The total area of the solar panels is modeled by the expression 69x²+680x,  where x is the width, in meters, of each solar panel.   

Questions to Solve  

Question 3 

The perimeters of the two figures are equal. Which expression represents the missing side length?    

Solution: 

Let the missing side length be m.  

Perimeter of parallelogram = 3x – 4 + 2x + 3x – 4 + 2x = 10x – 8  

Perimeter of triangle = 2x + 6 + 2x + 6 + m = 4x + m + 12  

Both are equal.  

10x – 8 = 4x + m + 12  

6x = m + 20  

m = 6x – 20 

Key Concepts Covered  

1. Standard form of a Polynomial 

2. Polynomials can be named according to the number of terms and their degree. 

3. Polynomial Operations 

Exercise

1. How does adding or subtracting polynomials compare to adding or subtracting integers?
2. Describe the relation between the degree of a monomial and the standard form of the polynomial.
3. Explain why the sum of x + x is 2x instead of x².
4. Rebecca says that all monomials with the same degree are like terms. Explain Rebecca’s error.
5. How is it possible that the sum of two quadratic trinomials is linear binomial?
   • Describe each statement as always, sometimes or never true.
6. A linear binomial has a degree of 0.
7. A trinomial has a degree of 2.
8. A constant has a degree of 1.
9. A cubic monomial has a degree of 3.
10. Combine like terms to simplify. (5x + 2) + (4x + 5)