Explainer: Polynomial Expressions

In this explainer, we will learn how to solve word problems through forming and evaluating polynomial expressions using the information provided in the questions.

In mathematics, we are often asked questions where we need to bring together various different skills that we have learned to solve a problem. For example, we may be asked a question that involves finding the area of a rectangle, but rather than being given the numeric lengths, we are given algebraic lengths; then, we are asked to form an expression for the area rather than calculate its value. Equally, we could be given a story problem for which we can form an equation as part of the steps for finding a solution. Answering these types of questions can be an excellent way to assess your understanding of a topic or improve your algebraic fluency.

Let us have a look at a series of examples in which we need to use our skills in algebra to write and evaluate polynomial expressions.

Example 1: Writing and Evaluating Polynomial Expressions

Write, in the simplest form, an expression that represents the perimeter of the given triangle.

Answer

First, recall that the perimeter of a triangle is the sum of the lengths of its sides. Therefore, an expression for the perimeter of the triangle is (5𝑥)+(5𝑥+8)+(6𝑥2).

Using the commutative and associative properties of addition, we can reorder the terms of the expression and regroup any like terms: (5𝑥+5𝑥+6𝑥)+(82).

Simplifying, we get our expression for the perimeter of the triangle as 16𝑥+6.

At this point, if we were told an explicit value of 𝑥 we could use our expression to calculate the perimeter of the triangle for that specific value.

Example 2: Writing and Evaluating Polynomial Expressions

Write an expression for the area for the shaded region in the shape below.

Answer

To answer this question, we need to calculate an expression for the area of the large rectangle and then subtract each of the expressions for the smaller rectangles which are unshaded. Notice that the two rectangles at the bottom are congruent squares and, therefore, have the same area. Recall that the area of a rectangle is calculated by multiplying the length and the width. That is, 𝐴=𝑙×𝑤.

An expression for the area of the large rectangle is (11𝑤+13)(13𝑤).

If we then expand the bracket using the distributive law, we have that this is equal to 143𝑤+169𝑤.

The area of the smaller rectangle at the top is (3𝑤)(2)=6𝑤, and the area of the two rectangles at the bottom is 2(3𝑤)(3𝑤)=18𝑤.

The area of the shaded region is, therefore, 143𝑤+169𝑤6𝑤18𝑤.

Using the commutative property along with the associative property, we can rewrite this as 143𝑤18𝑤+(169𝑤6𝑤), which simplifies to 125𝑤+163𝑤.

Let us move on to looking at some story problems that involve the formation and evaluation of different polynomial expressions. With these examples, we need to read the questions carefully to extract all the relevant information from the question that is needed to solve the associated problem.

Example 3: Writing and Evaluating Polynomial Expressions

Nine friends went to an amusement park. The cost of each ticket was $𝑥. The given table shows the prices of snacks and beverages at the park. Four of them bought a large pretzel and a bottle of water each. The remaining five did not buy anything. Write an expression that represents the total cost of tickets and snacks or beverages, and then simplify it.

Snack or BeverageLarge popcornLarge pretzelSmall sodaBottle of water
Price$7$4$2$3

Answer

We are told that nine friends went to an amusement park and each had to pay an entrance fee of $𝑥. We are then told that four of them bought a large pretzel and a bottle of water and the remaining five did not buy anything. We can, therefore, work out that the total paid by the four who bought snacks was 4(𝑥+4+3), and the remaining five spent a total of 5𝑥.

If we add these two results, we get an expression for the total amount of money spent in the amusement park, which is 4(𝑥+4+3)+5𝑥.

If we expand the parentheses and simplify, we get 9𝑥+28.

Alternatively, we could have found the total amount spent on tickets to be 9𝑥 and the total amount spent on snacks to be 4(4+3)=28, which, when summed, would arrive at the same simplified expression: 9𝑥+28.

Example 4: Writing and Evaluating Polynomial Expressions

Amelia received $75 on her birthday to buy a gift. She bought three new pairs of pants for 𝑠 dollars each. Write an expression to represent the amount of money she has left. Given that each pair costs $21, how much money does she have left?

Answer

Here, we are told that Amelia had a total of $75 and bought three pairs of pants for $𝑠 each. We are told to calculate an expression for the amount of money which she had left. This will be her total amount of money minus the amount she spent on pants, that is, 753𝑠.

We are then told that each pair of pants cost Amelia $21 and we are asked to calculate how much she had left. We can calculate this by substituting 𝑠=21 into our expression: 753(21)=12.

So, the answer is $12.

Let us finish by looking at one further example.

Example 5: Writing and Evaluating Polynomial Expressions

A car rental charges $25 a day in addition to $0.15 per mile. Write an expression that represents the cost of renting a car for one day to drive 𝑚 miles, and then determine the rental cost of a car after covering 46 miles.

Answer

Firstly, we are told that it costs $25 to rent a car for a day with an additional charge of $0.15 for every mile driven. And we are told to write an expression to represent the cost of renting a car for a single day and driving a total of 𝑚 miles. Our expression would, therefore, be 25+0.15𝑚.

We are then asked to calculate the cost of renting a car, for a single day, which is driven 46 miles. To do this, we can substitute 𝑚=46 into our expression and then evaluate: 25+0.15(46)=31.9.

This represents a cost of $31.90.

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