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Wize High School Grade 11 Math Textbook > Operations with Polynomials

Adding and Subtracting Polynomials

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Adding or Subtracting Polynomials



Similar Terms

Terms are considered similar if they have the same variables and those variables are raised to the same power.

Example 1

The following terms are considered similar terms.

1. 3x3and7x33x^3 \quad \text{and} \quad 7x^3

2. 2xy2and−3xy22xy^2 \quad \text{and} \quad -3xy^2

3. 5qand1q5q \quad \text{and} \quad 1q

Wize Tip
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 the 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 subtract 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)=x2+5x−1.05x2−0.5x−10P(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) &= \colorOne{x^2} \colorTwo{+ 5x} \colorOne{- 1.05x^2} \colorTwo{- 0.5x} - 10 \\ P(x) &= \colorOne{-0.05x^2} \colorTwo{+ 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 a fraction of a cup.

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 rectangular garden has sides that are three times as long as it is wide.

A small plot in the shape of a right isosceles triangle is going to be added to the side, as shown above.

Create a function F(x)F(x) that will represent the total area of the new garden.