A polynomial is an expression which consists of two or more than two algebraic expressions.  In a polynomial expression, the same variable has different powers. If the polynomial is added to another polynomial, the resulting expression is also a polynomial. The same goes with the operations of addition, subtraction, multiplication and division.

In this article, we will see how to find the unknown constants, and how to multiply and divide the polynomials. Below are some of the examples of polynomial word problems which you will find quite useful in understanding polynomials and their attributes when they are added, subtracted, multiplied or divided.

Exercise 1

Find a and b if the polynomial x^ 5 - ax + b is divisible by x ^2 - 4.

Exercise 2

Determine the value of m if 3x^2 + mx + 4 has x = 1 as one of its roots.

Exercise 3

Find a fourth degree polynomial that is divisible by x ^2 - 4 and is annuled by x = 3 and x = 5.

Exercise 4

Calculate the value of a for which the polynomial x ^ 3 - ax + 8 has the root x = -2. Also, calculate the other roots of the polynomial.

Exercise 5

The length of the rectangle is 2x - 3 and its width is equal to x - 1. Find the area of the rectangle.

Exercise 6

The number of tablets sold by a shop can be modeled by the expression N (t) = 7t + 25 and price per tablet is modeled by an expression P(t) = 3 t ^ 2 +3t +36, where t is the number of months in a year. If we use this model, what is the total amount of revenue generated by the shop at the end of the year?

Exercise 7

The area of the rectangle is given by the polynomial expression x ^ 3 - 2 x ^ 2 -6x + 12 and its length given by x - 2. Find the width of the rectangle.

Exercise 8

The distance covered by a bike is given by the expression 2x ^2 +6x -20. The time taken by the bike to covered this distance is given by the expression x - 2. Find the speed of the bike.

Exercise 9

The number of shirts sold by the shopkeeper is given by the expression 3x - 5. The price per shirt is given by the expression 2x + 1. Find the total amount of revenue earned by the shopkeeper by selling the shirts.

Solution of exercise Solved Polynomial Word Problems

Solution of exercise 1

Find a and b if the polynomial x^5 - ax + b is divisible by x^2 - 4.

Step 1

First, find factors of the expression x^2- 4. Since it is a perfect square, hence it can be written as:

x^2 - 4 = (x +2) \cdot (x - 2)

Step 2

Set the factors equal to zero:

Either (x+2) =0 or (x - 2)=0

Hence, x = -2 or x = 2

Step 3

Put these values of x in the original polynomial expression:

a) Substitute x = -2

P(-2) = (-2)^5 − a \cdot (-2) + b = 0

-32 + 2a + b = 0

Take -32 on the right hand side of the equation:

2a +b = 32

b) Substitute x = 2

  P(2) = 2^5 - a  \cdot 2 + b = 0

32 - 2a +b = 0

− 2a +b = -32

Step 4

Add both expressions together to get

2a + b - 32 - 2a + b + 32 = 0

2b = 0

Hence, b=0

Step 5

Remember we got the expression 2a + b = 32 in the above problem. To find the value of a, put b = 0 in this expression:

2a + 0 = 32

Divide both sides by 2 to get the value of a:

a = 16

 

Solution of exercise 2

Determine the value of m if 3x ^ 2 + mx + 4 has x = 1 as one of its roots.

Step 1

Put x = 1 in the original polynomial expression:

P(1) = 3 \cdot 1 ^2 +m \cdot 1 + 4 = 0

3 + m + 4 = 0

Step 2

Take 4 on the left side of the equation:

3 + m = -4

Subtract 3 from both sides of the equation to get the final answer:

m = -7

Solution of exercise 3

Find a fourth degree polynomial that is divisible by x ^ 2 − 4 and has the roots by x = 3 and x = 5.

Step 1

x = 3 and x = 5 can be written as x - 3 = 0 and x - 5 =0

Step 2

Multiply x - 3 and x - 5 together:

(x - 3) \cdot (x - 5)

x (x - 5) -3 (x - 5)

x ^ 2 -5x -3x +15

x ^2 -8x +15

Step 3

Now, multiply x ^2 -8x +15 with x ^ 2 − 4 to get the fourth degree polynomial:

= <span class="actividades_v">x ^ 2 − 4 \cdot  x ^2 -8x +15

=x ^ 2 (<span class="actividades_v">x ^2 -8x +15) - 4 (x ^2 -8x +15)

=x ^ 4 - 8x ^3 +15x ^2 -4x ^2 +32x  - 60

= x^4 - 8x^3 + 11x ^ 2 +32x - 60

Solution of exercise 4

Calculate the value of a for which the polynomial x ^ 3 − ax + 8 has the root x = −2

Put x = -2 in the polynomial expression:

P(-2) = (-2) ^ 3 - a \cdot (-2) + 8 = 0

-8 + 2a +8 = 0

   <span class="sol">a= 0

Solution of Exercise 5

Given that the length is 2x - 3 and width is x - 1.

The area of the rectangle = length x width

= (2x - 3)(x - 1)

=2x (x -1) - 3 (x - 1)

=2x ^2 -2x -3x +3

=2x ^2 -5x +3

Hence, the area of the rectangle is 2x ^2 -5x +3.

Solution to exercise 6

We know that the amount of revenue generated is equal to the:

Number of items sold x Price per item

Here, Number of items sold N (t) = 7t + 25

Price per tablet P(t) = 3 t ^ 2 +3t +36

Multiply these two expressions together:

(7t + 25) \cdot (3 t ^2 +3t +36)

7t ( 3 t ^2 +3t +36) + 25 (3 t ^2 +3t +36)

21 t ^3 +21 t ^2 +252t +75t ^2 +75t+900

21 t ^ 3 + 96 t ^2 +327 t +900

Put t = 12 in the above expression because in a year there are 12 months:

21 {12} ^ 3 + 96 {12} ^2 +327 (12) +900

36288 + 13824 + 3924 + 900

54936

Hence, the total revenue of the shop for a year is 54936 dollars.

Solution to exercise 7

The area of the rectangle = x ^ 3 - 2 x ^ 2 -6x + 12

Length of the rectangle = x - 2

The formula for area of the rectangle = length x width

Hence, to find the width of the rectangle, we need to divide the area by the length:

\frac {x ^ 3 - 2 x ^ 2 -6x + 12}{x - 2}

Use the polynomial long division method to solve the above expression:

 

\begin{array}{ccccccccccc} &&&  x^2 & + & 6 &\\ \cline{3-10} \multicolumn{2}{r}{x - 2 \surd} & x^3 &-& 2x^2  &-& 6x &+&  12 & \\ & &x^3&-& 2x ^ 2&& & & & \\ \cline{3-7} & & & & &-& 6x&+& 12 & & & & & \\ & & & & & -& 6x&+& 12& & && & \\ \cline{4-8} & & & && && 0 \\ \end{array}

The quotient is the width of the rectangle. Hence, the width of the rectangle = x ^ 2 + 6

 

Solution to exercise 8

Since the formula for the distance is speed x time, hence we can easily derive formula of speed from this formula of distance:

speed = \frac {distance}{time}

Put the values in the questions in the above formula to get the speed:

speed = \frac {2x ^2 +6x -20}{x - 2}

Use the polynomial long division method to find the answer.

\begin{array}{ccccccccccc} &&&  2x & + & 10 & \\ \cline{3-10} \multicolumn{2}{r}{x - 2 \surd} & 2x^2 &+& 6x  &-& 20 & \\ && 2x^2&-& 4x && & & & \\ \cline{3-7} & & & & 10x&-& 20& & & & & \\ & & & & 10x&-& 20& & && & \\ \cline{4-8} & & & && && 0 \\ \end{array}

Hence, the speed of the bike is 2x + 10.

Solution to exercise 9

The total amount of profit is calculated by the formula:

Profit = Price per item x Number of items sold

Hence, we will find the profit by multiplying the price of the single shirt with the total number of shirts sold.

Price of a single shirt = 2x + 1

Number of shirts sold = 3x - 5

Profit = (2x + 1)(3x - 5)

= 2x (3x - 5) + 1(3x - 5)

= 6x ^2 -10x +3x - 5

= 6x ^ 2 -7x - 5

Hence, the total profit earned by the shopkeeper = 6x ^ 2 -7x - 5

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.

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