# Algebra: Polynomials Word Problems

This page is for algebra students who need help, and for teachers and tutors who are looking for practice word problems on polynomials.

### Algebra

Polynomials in algebra.

### Pre-Calculus

Polynomials in pre-calculus.

### Pre-Algebra

Polynomials in pre-algebra.

### Division Of Polynomials

Division Of Polynomials in algebra.

### Multiplication Of Polynomials

Multiplication Of Polynomials in algebra.

### Subtraction Of Polynomials

Subtraction Of Polynomials in algebra.

### Operations With Polynomials

Operations With Polynomials in algebra.

### Factoring Polynomials

Factoring Polynomials in algebra.

### Monomials

Monomials in algebra.

### Trinomials

Trinomials in algebra.

### Polynomial Functions

Polynomial Functions in algebra.

### Rational Zeros

Rational Zeros in algebra.

### Binomials

Binomials in algebra.

### Long Polynomial Division

Long Polynomial Division in algebra.

# Algebra: Polynomials Word Problems

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# Algebra: Polynomials WordProblems

A box with no top is to be made from a piece of metal
A box with no top is to be made from an 8 inch by 6 inch piece of metal by cutting identical squares from each corner and turning up the sides. The volume of the box is modeled by the polynomial 4x3-28x2+48x. Factor the polynomial ...
Long Polynomial Division
How is dividing a polynomial by a binomial similar to or different from the long division you learned in elementary school?
Suppose you are at the gas station
Suppose you are at the gas station filling your tank with gas. The function C(g) represents the cost C of filling up the gas tank with g gallons. Given the equation:
C(g)=3.03(g)
a) What does the number 3.03 represent?
b) Find C(2)
c) Find C(9)
d) For the average motorist, name one value for g that would be inappropriate for this function’s purpose. Explain why you chose the number you did.
e) If you were to graph C(g), what would be an appropriate domain? ...
The mainsail of a lightning sailboat
The mainsail of a lightning sailboat is a right triangle in which the hypotenuse is called the leech. If a 24 ft tall mainsail has a leech length of 26ft and if Dacron sailcloth casts \$10 per square ...
Find the remainder when the polynomial
Find the remainder when the polynomial -x3+7x-6=0 is divided by x-1
The sum of the squares of the two
The sum of the squares of the two larger of three consecutive even integers is 12 less than 4 times the square of the smaller ...
The area of a rectangle is given
The area of a rectangle is given by the following polynomial 44t-2t2...
A rectangular garden is 5 feet longer
A rectangular garden is 5 feet longer than twice its width. It has a sidewalk 3 feet wide on two of its sides. The area of the sidewalk is 213 square ...
A homeowner is installing a swimming pool
A homeowner is installing a swimming pool in his backyard. He wants its length to be 4 feet longer than its width. Then he wants to surround it with a concrete walkway 3 feet wide. If he can only afford 300 square feet of concrete for the ...
A rectangular picture frame
A rectangular picture frame is 4 times as wide as its length with a perimeter of 80 ...
A box has a volume of 1,800 cubic cm
A box has a volume of 1,800 cubic cm. The length is 6 times the height. As well, the length is 18 cm longer than the width. If the length is increased by 2 cm, the width by 3 cm and the height by 4 cm, the volume becomes 4,320 cubic ...
polynomial functions
Four boys are playing with their marbles. Two boys have the same number of marbles and they both have 1 more marble than the 3rd boy. The 4th boy has 1 marble less than the 3rd boy. If all the number of marbles of each boy is multiplied, the product would be 18...
Jurassic Park charges \$20 for adults
Jurassic Park charges \$20 for adults and \$12 for children. Total bill for 454 people was \$3860...
Kattya has a square garden
Kattya has a square garden. When she increases her garden's width by 3 meters and decreases its length by 4 meters, her garden becomes 60 m2...
When a certain polynomial is divided
When a certain polynomial is divided by x + 2, the quotient is x2 -4x + 1 and the remainder is 8...
ABCD is a rectangle
ABCD is a rectangle
AC = 4X +6  BD = 5X-2
Find the length of each diagonal.
Quotient
The quotient of two numbers is 4 and their difference is 39...
The area of a rectangle
The area of a rectangle is 54cm2. The length is 3 cm less than twice the ...
A company produces a product
A company produces a product for \$9.45 per unit, plus a fixed cost of \$5200. The company sells the product for \$15.80 per unit.
a) determine a function, c(x), to represent the cost production x units
b) determine  a function, r(x) to represent the revenue from selling x units
c) determine a fucntion to represent profit
d) caculate break even point
Find 3 consecutive even numbers
Find 3 consecutive even numbers where the product of the smaller two numbers is 64 less than the square of the largest number.
Given a trapezoid where the small base
Given a trapezoid where the small base is 6 greater than the height and the larger base is 4 greater then three times the height, find all three dimensions (small base, large ...
Solution A is 10% alcohol
Solution A is 10% alcohol. Solution B is 60% ...
The length of a rectangle
The length of a rectangle is 4 less than 3 times its width. Represent the area for this rectangle
A rectangular shipping container
A rectangular shipping container has a volume of 2500 cm3. The container is 4 times as wide as it is deep, and 5cm taller than it is ...
A triangle has a perimeter
A triangle has a perimeter of 197 centimeters. The sum of the smallest and medium sides is 49 cm more than the longest side. The difference of the longest side and shortest sides is 22 ...
Find two consecutive odd integers
Find two consecutive odd integers such that the square of the larger is 3 less than 12 times the smaller
Linda has a garden
Linda has a garden. The width of her garden is 8 feet longer than the length of it. Around the garden she has a 4 foot wide sidewalk. The area of the sidewalk is 320 feet ...
The product of two consecutive odd numbers
The product of two consecutive odd numbers is 195...
A diver stands at the very end of a diving board
A diver stands at the very end of a diving board before beginning a dive. The deflecton (d) of the board at a position (s) feet from the stationary end is given by d=cs2(3L-s) for 0 is greater than or equal to s is greater than or equal to L, where L is the length of the board and c is a positive constant that depends on the weight of the diver and on the physica properties of the board. Suppose the board is 10 ft long.

a) If the deflection at the end of the board is 1 ft, find c
b) Show that the deflection is .5 foot somewhere between s=6.5 and s=6.6
A rectangular box is (2x + 7) feet wide
A rectangular box is (2x + 7) feet wide, (5x + 2) feet long, and (x - 4) feet ...
The square of the largest of three consecutive integers
The square of the largest of three consecutive integers is 320 less than the sum of the squares of the two smaller ...
The lenght of a rectangle
The lenght of a rectangle is 30cm less then three times the width. the perimeter of this rectangle is 180...
In a retail store, a fake diamond ring
In a retail store, a fake diamond ring cost \$10 dollars. Joan buys two rings, and pays the cashier with a \$2x bill. Write the expression that represents the change she would receive in polynomial form
The Robertsons put a rectangular pool
The Robertsons put a rectangular pool with a stone walkway around it in their back yard. The total length of the pool and walkway is 3 times the total width. The walkway is 2ft. wide all around.

a.) Write an expression for the area of the pool.
b.) Find the area of the pool when the total width is 10ft.
...
A catering company is designing a box
A catering company is designing a box. The volume box is to be 54 cubic inches and the bottom of the box to be a square. Suppose the bottom of the box has a width that is 3 inches smaller than the height x of the ...
A circular pond will be placed
A circular pond will be placed on a square piece of land. The length of a side of the square is 2x. The radius of the pond is x. The part of the square not covered by the pond will be planted with flowers. What is the area of the region that will be planted with ...
A rectangle has a width
A rectangle has a width of 3x+5, the length is twice ...
Polynomials
(36b3+18b+44b+49)  / (6b+5)
Together Steve and Heather`s ages
Together Steve and Heather's ages add up to be ten. If we let "x" be Steve's ...
A photo is 4 inches longer
A photo is 4 inches longer than it is wide. A 3-inch border is placed around the photo making the total area of the photo and border 165 square ...
Greenshield`s Formula can be used
Greenshield's Formula can be used to determine the amount of time a traffic light at an intersection should remain green.  G = 2.1n + 3.7   G= green time in seconds   n = average number of vehicles traveling in each lane per light cycle. Write a simplified expression to represent the average green light time per vehicle
A polynomial models the profit a company makes
The polynomial
s3 - 70s2 + 1500s - 10800
models the profit a company makes on selling an item at price ...
Three students in an outdoor pursuits program
Three students in an outdoor pursuits program collaborated to write an adventure travel guidebook for Alberta and BC. Two local publishers are interested in marketing and publishing it and both anticipate retailing the book for \$20. Publisher "A" offers a straight royalty of 10% (on the retail price) on all book sales. Publisher "B" offers an 8% royalty for any books sold up to 5000 copies and will then increase the royalty to 15% for any copies sold in excess of 5000.

a) Describe the type of function
b) Find the functions A(x) and B(x) - notice that function A(x) is apiecewise defined function.
c) Let x be the number of books sold and let A(x) give the income under plan "A" and B(x) give the income under plan "B". Graph functions on the interval 0,10000
A cuts half a rectangular lawn
A cuts half a rectangular lawn, 50 m by 20 m by mowing strips of equal width around the perimeter. B cuts the small rectangle left. How wide a strip does A cut so that A and B share the work equally??
You are designing a rectangular flower bed
You are designing a rectangular flower bed that you will border using bricks pavers. The width of the border around the bed will be the same on every side.
a. Write a polynomial that represents the total area of the flower bed and the border.
b. Find the total area of the flower bed and border when the width of the border is 1...
A rectangular parking lot
A rectangular parking lot is 50 ft longer than it is ...
A manufacturer of screened porches
A manufacturer of screened porches sells a model that sells for p dollars. The revenue from the sale of this model is given by the following polynomial:

R = -0.25p2 + 250p

If revenue is found by multiplying price by ...
A polynomial in x has degree 3
A polynomial in x has degree 3. The coefficient of x2 is 3 less than the coefficient of x3. Te coefficient of x is three times the coefficient of x2. The remaining coefficient is more than the coefficient of x3. The sum of the coefficients is -4...
A market research finds that at a price
A market research finds that at a price of p=20, they would sell x=42 tiles each month. If they up the price to p=10, then more people would purchase the tile, and they can expect to sell x=52 tiles a months ...
A quilt patch cut in the shape
A quilt patch cut in the shape of a triangle has a base of 5x inches and a height of 1.732x ...
A square has sides of length
A square has sides of length 3x+1 meters. Find a polynomial A(x) that represents the area of the square. Find A(1)
A rectangle is 15m longer than it is wide
A rectangle is 15m longer than it is wide. If both its length and width are increased by 5 m and its area is increased by 200 ...
polynomials
Superman wants to build a sidewalk around a lot. Batman says that the current width of the lot is 5 meters and the length is 7 meters. Spiderman says that the final area must be 63 square meters. And Aquaman says that the width of the walkway is X meters ...
A water balloon is released from a window
A water balloon is released from a window 96 feet above the ground. The formula h=-16t2+16t+96 describes the height of the water balloon above the ground, h, in feet, t seconds after the ...
At p dollars per ticket
Total Revenue
At p dollars per ticket, a promoter expects to sell 40,000 -1000p tickets to a concert.
a) How many tickets will she sell at \$10 each?
b) At \$10 per ticket, what is the total revenue?
c) Find a polynomial R(p) that represents the total revenue when tickets are p dollars each.
d) Find R(20), R(30), and R(35)
A television video game company has the following total expenses
A television video game company has the following total expenses E and total incomes I for producing x number of ...
Find a polynomial for the perimeter and area
Find a polynomial for the perimeter and area of a rectangle d +2 and d
A red ball and a green ball are simultaneously tossed
A red ball and a green ball are simultaneously tossed into the air. The red ball is given an initial velocity of 96 feet per second, and its height t seconds after it is tossed is feet. The green ball is given an initial velocity of 80 feet per second, and its height t seconds after it is tossed is ...
Find a polynomial A(x) that represents the area
Find a polynomial A(x) that represents the area of the ...
A stage is rectangular in shape
A stage is rectangular in shape. The width is 10ft less then the length. If the area of the stage is 600...
A rectangular yard has a length
A rectangular yard has a length 7 feet greater than it's width. If both dimensions are increased by 8 ...
The volume of one of the buildings
The volume of one of the buildings in the downtown area is 839,808 cubic meters. The building is 9 times a s tall as the radio tower on top of the building. The square base has a side that is 27 times 4 meters less than the height of the radio ...
The revenue for selling y units
The revenue for selling y units is
R = 3y2 – 2y + 5
and the cost of producing y units is
C = y2 + y – 3...
Alex, Bill and Danny collect tropical fish
Alex, Bill and Danny collect tropical fish. Alex has 8 fewer than Danny and Bill has 2 more than ...
A truck and a car leave an intersection
A truck and a car leave an intersection. The truck travels south and the car travels east. When the truck had gone 24 miles, the distance between the car and the truck was four miles more than three times the distance traveled by the car heading east. Suppose the truck stops at point c and the car stops at point ...
From 1992 to 2003, the number of people in the United States
From 1992 to 2003, the number of people in the United States who participated in snowboarding can be modeled by
f(t) =0.013t4-0.021t3+0.084t2+0.037t+1.2
where f(t) is the number of participants ( in millions) and t is the number of year since 1992...
The world`s largest swimming pool
The world's largest swimming pool is the Orthlieb Pool in Casablanca,Morocco. The length is 30 meters longer than 6 times the width. If the perimeter of the pool is 1,110 ...
If a vendor charges P dollars
If a vendor charges P dollars each for rugby shirts, then he expects to sell 2000-100P shirts at a tournament.
A) Find a polynomial R(P) that represents the total revenue when the shirts are P dollars.
B) Find R(5), R(10)...
Antonio is preparing to make
Antonio is preparing to make an ice sculpture. He has a block of ice that he wants to reduce in size by shaving off the same amount from the length (3ft.), width (5ft.), and height (4ft.). He wants to reduce the volume of the ice block to 24 cubic ...
You are making a srapbook
You are making a srapbook out of a picture that are 7 inches long and 5 inches wide. you want to place a border of equal width on each ...
Calculate the perimeter of the rectangle
Calculate the perimeter of the rectangle if the area of the polynomial is 6x2+17x+5
Two rectangles have the same width
Two rectangles have the same width. The length of one is 2 feet longer than the width. The length of the other is 4 feet longer than the width. The larger rectangle has 8 more square feet than the ...
A triangle is 10 cm wider than it is tall
A triangle is 10 cm wider than it is tall. The area is 28cm2...
Two rectangles have the same width
Two rectangles have the same width. The length of one is 1 foot longer than the width. The length of the other is 2 feet longer than the width. The larger rectangle has 4 more square feet than the ...
If the ticket agency charges
If the ticket agency charges "p" dollars per ticket for a concert, it can sell maximum of (40000 – p) tickets. Find:
a. A polynomial to represent the total revenue with a ticket price as p dollars.
b. How many tickets will be sold for a price of \$10 each.
...
The owner of a daycare center plans to have a rectangular sandbox
The owner of a daycare center plans to have a rectangular sandbox built in the outdoor play area. The width (in feet) of the sandbox is given by the polynomial 2x-5. If the length of the sandbox is designed to be 10 feet longer than the width, write a polynomial that represents the area of the sandbox.
(1)  Part A: Solve the above problem using any methods, concepts, and procedures learned in this module. All work must be shown and completed in an equation editor.
(2)  Part B: Write a 250 word summary of the above problem. Your written work should include but necessarily be limited to the following; an explanation of the problem, a review of the mathematical concepts involved, a critique of the method used to obtain the solution, and an interpretation of the answer. Your summary should be written in APA format, demonstrate proper use of the English ...
Polynomials
The width of a rectangle is 4 feet less than the lengh. The area is 5ft2...
Find 3 consecutive even numbers
Find 3 consecutive even numbers where the product of the smaller two numbers is 52 less than the square of the largest number.
Consider the function
Consider the function f(x) = x3 + 4x2 + kx - 4
The remainder from f(x) divided by (x+2) twice the remainder from f(x) divided by (x-2)...
A triangle has sides
A triangle has sides 2x -5, 3x + 1, and 4x + 2...
A ruler orders his chamberlain to collect
A ruler orders his chamberlain to collect an army from 30 houses, in such a way that he takes the same number of men from each house as he has already collected. The servant goes to the first house alone; to the second house he goes with one other and so ...
James is 5 years older than Jennifer
James is 5 years older than Jennifer. Three years ago the product of their ages was 36 less than the product of their ...
Choose the polynomial that represents
Choose the polynomial that represents the area of the rectangle with sides (x+8) and (x-1)
Polynomial representation of the width of a rectangle
A(x)=x3-1 the length is x-1
A) Find the polynomial to represent the width of the ...
Pure acid is to be added
Pure acid is to be added to a 5% acid solution to obtain 95L of 23% solution. What amounts of each should be used?

How many liters of 100% pure acid should be used to make the ...
Two rectangles have same width
Two rectangles have same width. The length of one is 1 foot longer than the width. The length of the other is 2 feet longer than the width. The larger rectangle has 5 more square feet than the ...
The area of a rug is 36 square feet
The area of a rug is 36 square feet. The length of the rug is 5 feet longer than the ...
The length of a rectangle
The length of a rectangle is 2 inches more than 3 times the ...
The area of a rectangle
The area of a rectangle is 2x2 +7x -9 and the width is x-1...
Vanessa built a rectangular pen for her dogs
Vanessa built a rectangular pen for her dogs. She used an outside wall of the garage for one of the sides of the pen. She had to buy 20 m of fencing in order to build the other sides of the ...
A triangle has sides
A triangle has sides 2x – 5, 3x + 1, and 4x + 2...
The length is 30 meters longer
The length is 30 meters longer than 6 times the width. If the perimeter of the pool is 1,110 ...
Find the polynominal for the rectangle
Find the polynominal for the rectangle with the perimeter c+6 and for the area c.
The length of a rectangle
The length of a rectangle is x+5 and the width is x-2...
The length of a rectangle
The length of a rectangle is three inches greater than the width.
What is the area of a square
What is the area of a square measuring (5a - 2b) cm each side ?
The area of a rectangle
The area of a rectangle is 156 ft. The length is 4ft greater than the ...
If a cylinder with a height of 3 inches
If a cylinder with a height of 3 inches and a raduis of 2 inches is compared to a cube with 3 inche edges which expression represents the volume of the cylinder to the volume of the cube?
Find 4 consecutive odd integers
Find 4 consecutive odd integers where the product of the two smaller numbers is 64 less than the product of the two larger numbers.
The base of a triangle is 3 cm greater
The base of a triangle is 3 cm greater than the height. The area is 44...
Colin has a 500 cm ribbon
Colin has a 500 cm ribbon that is cut into 3 pieces. The second piece is 3 times longer than the first. The third piece is 4 cm less than 4 times the length of the ...
Suppose the market research company find
Suppose the market research company find that at a price of p=\$20, they would sell d=42 dresses each month. If the price was lower like p= \$10 more people would purchase, and they expect to sell d=52 dresses in a month's time. Find the equation of the line for the demand ...
Find the polynomial that repersents
Find the polynomial that repersents the volume of a cube if each side measures x+6
The minute hand of a click is 3 cm longer
The minute hand of a click is 3 cm longer than the hour hand. If the distance between the tips of the hands at 9 o'clock  is 15 ...
The length of a rectangle
The length of a rectangle is 5 centimeters less than twice its width the perimeter of the rectangle is 26cm what are the dimensions of the retangle?
The square of the largest of the three consecutive integers
The square of the largest of the three consecutive integers is 320 less than the sum of the squares of the two smaller integers. Find the three integers
If two numbers differ
If two numbers differ by 7, then what polynomial represents their ...
Find the area of the rectangle
Find the area of the rectangle when the width is 10 centimeters.
Polynomials
Find the dimensions of a rectangle whose perimeter is 46m and whose area is 104m2.
The measure of the area of a rectangle
The measure of the area of a rectangle is 4m2-3mp+3p-4m. If the dimensions of the rectangle are represented by polynomials with integral ...
The length of a rectangular swimming pool
The length of a rectangular swimming pool is 2x-1 meters and the width is x + 2 meters. Write a polynomial A(x) that represents the area. Find A(5)
The width in feet of the sandbox
The width in feet of the sandbox is given by the polynomial 2x-5. If the length of the sandbox is designed to be 10 feet longer than the ...
The edge of one cube is 3 shorter
The edge of one cube is 3 shorter than the edge of a second cube. The volumes of the two cubes differ by 513cm3...
The difference of two positive numbers
The difference of two positive numbers is 5 and the sum of thier squares is 233...
The length of the pool is 30 meters longer
The length of the pool is 30 meters longer than 6 times the width. The perimeter of the pool is 1110 ...
Write an expression for the area
Write an expression for the area of a 1 foot border around a circular pool with a radius of r feet.
The length of a rectangle
The length of a rectangle is 3 inches greater than the ...
Suppose that P dollars is invested in a savings account
Suppose that P dollars is invested in a savings account at interest rate i, compounded annually, for 2 years. The amount A in the account after 2 years is given by:
A=P(1+i)2
Find the volume of a cylinder
Find the volume of a cylinder with radius 3b and height bc.
The volume of a rectangular prism
The volume of a rectangular prism is (15r3-17r2-42r). If the dimensions of the prism are represented by polynomials with integral ...
The perimeter of a rectangle
The perimeter of a rectangle is 140m. The length is 4m more than twice the ...
The area of a rectangle
The area of a rectangle is 165 ft squared. The length is 4 ft greater than the ...
The smaller of two numbers
The smaller of two numbers is 64 less than the square of the largest number.
The length of a triangle is twice the width
The length of a triangle is twice the width. A square is cut out at the center. The remaining area is 63 square ...
The volume of a right circular cylinder
The volume of a right circular cylinder is 12 in. Express the height as a function of the ...
The length of a rectangle
The length of a rectangle is 3 cm greater than the width. If each dimenstion is increased by 2 cm, the area is increased by 26 cm ...
The length of a rectangle is 8 cm less than 3 times the width
The length of a rectangle is 8 cm less than 3 times the ...
The volume of a box
The volume of a box with height x, length x +1, And width 2x - 4 is given by the trinomial 2x3 - 2 x2 - 4 ...
The hypotenuse of a right triangle
The hypotenuse of a right triangle is 25m long. The length of one leg is 10 m less than twice the ...
The width of a triangle is 3 feet less
The width of a triangle is 3 feet less than the length. The area is10ft2...
The length of a rectangle
The length of a rectangle is 8 times its width. If the length was decreased by 10m and the width decreased by 2m, the area would be decreased by 162m2...
The length of a rectangular board
The length of a rectangular board is 9 meters greater than the width. If the perimeter of the board is 26 ...
The area of a rectangle
The area of a rectangle is 165 ft2. The length is 4 ft greater than the ...
The perimeter of a rectangular swimming pool
The perimeter of a rectangular swimming pool is 42 meters. The length is 5 meters more than the ...
The length of a rectangle is increased
The length of a rectangle is increased by 20% and its width 50%...
The sides of a sandbox
The sides of a sandbox are 3x2 and x+8 feet wide.
a. What is the area of the sandbox?
b. If the depth of the sandbox is x ...
The length of a rectangle
The length of a rectangle is 11 cm more than its width. The area of the rectangle is 2040 cm2...

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