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Wize Grade 11 Mathematics Textbook > Polynomials and Operations

Adding and Subtracting Polynomials

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Adding or Subtracting Polynomials



Similar Terms

Terms are considered similar, or just similar terms, if they have the same variables and those variables are raised to the same powers.

Example 1

The following terms are considered similar terms.

1. 3x37x33x^3 \hspace{0.5cm}7x^3

2. 2xy2−3xy22xy^2\hspace{0.25cm}-3xy^2

3. 5q1q5q\hspace{0.5cm}1q

Wize Tip
Similar terms are also called "like" terms. When you hear the phrase, "combine like terms" it means combine all of the terms that are similar to one another.


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Adding or Subtracting Polynomials

To add or subtract polynomials their similar terms are combined.

Example 2

Simplify the following polynomials by combining like terms.

1. y=5x2−3x+2x2+7x+1y = 5x^2 - 3x + 2x^2 + 7x +1

y=5x2−3x+2x2+7x+1y=7x2+4x+1\begin{aligned} y &= \colorOne{5x^2} \colorTwo{- 3x} \colorOne{+ 2x^2} \colorTwo{+ 7x} + 1 \\ y &= 7x^2 + 4x + 1 \end{aligned}

2. y=3x2−2x−5x+x−3y = 3x^2 - 2x - 5x + x - 3

y=3x2−2x−5x+x−3y=3x2−6x−3\begin{aligned} y &= 3x^2 \colorOne{-2x - 5x + x} - 3 \\ y &= 3x^2 -6x - 3 \end{aligned}
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Example: Adding and Subtracting Polynomials




Jordan owns a small coffee shop. By carefully keeping track of receipts and expenses Jodan can model the total revenue and cost of their business. These are given by the functions R(x)R(x) and C(x)C(x)respectively, where xx represents the cups of coffee sold.

R(x)=x2+5x,C(x)=1.05x2+0.5x+10R(x) = x^2 + 5x, \hspace{0.5cm} C(x) =1.05x^2 + 0.5x + 10

To calculate the total profit, we can subtracting the cost away from the total revenue.

1. Write a function that expresses the total profit of Jordan's business.

P(x)=R(x)−C(x)P(x)=x2+5x−(1.05x2+0.5x+10)P(x)=−0.05x2+4.5x−10\begin{aligned} P(x) &= R(x) - C(x) \\ P(x) &= x^2 + 5x - (1.05x^2 + 0.5x + 10) \\ P(x) &= -0.05x^2 + 4.5x - 10 \end{aligned}

2. What is the domain of this function?

Since xx represents the number of cups of coffee sold, there is no way Jordan can sell a negative number of cups.
We would also assume that they can not sell fraction of cups.

Domain: non-negative integers

3. The value of this function can sometimes be positive or negative. How can we interpret this in the context of the problem?

When the profit function is positive the business is making money.
When the profit function is negative the business is losing money.

Practice: Adding and Subtracting Polynomials

Simplify the following polynomials by combing like terms.

1. p(x)=5x2+3x2−4x+2x−8p(x) = 5x^2 + 3x^2 - 4x + 2x - 8

2. f(x)=7x4−3yx4−8x4+y4f(x) = 7x^4 - 3yx^4 - 8x^4 + y^4

Practice: Adding and Subtracting Polynomials

Simplify the expression by combining all of the like terms

y=(3x3−5x+1)+(2x2+4x+1)−(7x3−x+1)y = (3x^3 - 5x + 1) + (2x^2 + 4x + 1) - (7x^3 - x + 1)

Practice: Adding and Subtracting Polynomials

A rectangle garden has sides that are three times as long as it is wide. The area can be described by the function A(x)A(x)

A(x)=3x2A(x) = 3x^2

A small plot in the shape of an isosceles triangle is going to be added to the side. Its area is given by B(x)B(x)

B(x)=12x2B(x) = \frac{1}{2}x^2
Create a function F(x)F(x) that will represent the total area of the new garden.