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Writing Polynomial Expressions from Situations

Turn word problems about areas, borders, boxes, numbers and revenue into polynomial expressions.

Practice this skill 25 min Write polynomial expressions from context Leer en español
Key idea

What you'll learn

Many GED questions describe a situation in words and ask, "Which expression represents…?" In earlier lessons you wrote linear expressions such as 3x + 5. In this lesson the situations produce polynomials: expressions with x^2 or x^3, usually because two expressions that both contain x get multiplied.

You will learn to:

  • Translate phrases like "the square of the sum" and "five less than the square" correctly.
  • Write area, perimeter and volume expressions for shapes whose sides contain a variable.
  • Write expressions for borders, walkways, frames and open boxes.
  • Write revenue expressions such as (price) × (number sold) when both change.
  • Simplify the expression by multiplying it out, and check it with a test number.

Why it matters on the GED. These "model it" questions are common in the algebra part of the test. The answer choices usually include several expressions that look alike: one correct model and others built from typical slips, such as adding instead of multiplying or adding the border to only one side. If you can build the expression step by step, you can pick the right one with confidence.

Real life. Contractors use these expressions to price a walkway around a pool. Businesses use them to see how a price change affects income. Packaging designers use them to see how cutting a box changes its volume.

Words to know

  • Variable — a letter that stands for an unknown number, such as x or w.
  • Polynomial — a sum of terms with whole-number exponents, such as 2w^2 + 3w or 4x^3 - 64x^2 + 240x.
  • Product — the result of multiplying. "The product of n and n + 2" is n(n + 2).
  • Sum / difference — the result of adding / subtracting.
  • Square of a number — the number times itself: the square of n is n^2.
  • Consecutive integers — whole numbers in a row: n, n + 1, n + 2. Consecutive even or odd integers go up by 2: n, n + 2, n + 4.
  • Area — the space inside a flat shape, in square units. Rectangle: length × width.
  • Perimeter — the distance around a shape. Rectangle: 2(\text{length}) + 2(\text{width}).
  • Volume — the space inside a solid, in cubic units. Box: length × width × height.
  • Uniform border — a strip of the same width on every side of a shape (a frame, a walkway, a margin).
  • Revenue — money taken in: price per item × number of items sold.
  • Simplified form (standard form) — like terms combined and written from highest power to lowest, such as -20x^2 + 60x + 3600.

All Math words in the glossary

Translating words into polynomial expressions

Most of the work is reading carefully. Translate one phrase at a time and keep the order the words tell you.

WordsExpressionNotes
the square of a numbern^2not 2n
five less than the square of a numbern^2 - 5"less than" flips the order
the square of five less than a number(n - 5)^2square the whole group
three times the square of a number3n^2only n is squared
the square of three times a number(3n)^2 = 9n^2the 3 is squared too
the product of a number and 4 more than itn(n + 4)multiply the two quantities
the product of two consecutive integersn(n + 1)
the product of two consecutive odd integersn(n + 2)odd numbers are 2 apart

"Less than" reverses order. "Seven less than x^2" means start with x^2 and take away 7: x^2 - 7. Writing 7 - x^2 is a classic trap.

Grouping words. "The square of the sum of x and 4" squares the whole sum: (x + 4)^2. "The sum of the square of x and 4" squares only x: x^2 + 4. Look at which noun comes right after "the square of."

Two phrases compared. Left: 'the square of the sum of x and 4' is a square with side x + 4, made of x squared, two 4x rectangles and 16, so (x + 4) squared = x squared + 8x + 16. Right: 'the sum of the square of x and 4' is just an x by x square plus 4 more, x squared + 4. The two areas are different.x²4x4x16x4x4(x + 4)² = x² + 8x + 16the square of the sumx²x+4x² + 4the sum of the square and 4

Simplify after you write it. Answer choices are often in simplified form, so multiply out:

n(n + 4) = n^2 + 4n \qquad (n - 5)^2 = n^2 - 10n + 25

Remember that (n - 5)^2 is (n - 5)(n - 5). It is not n^2 - 25 and not n^2 + 25.

Shapes whose sides contain a variable

When a side length is an expression, use the usual geometry formula and put each side in parentheses.

Rectangle. If the length is 3 feet more than twice the width w, then length = 2w + 3.

\text{Area} = w(2w + 3) = 2w^2 + 3w \qquad \text{Perimeter} = 2w + 2(2w + 3) = 6w + 6

Area has a w^2 because two expressions containing w are multiplied. Perimeter is only added, so it stays linear.

Square. Side x + 5: Area = (x + 5)^2 = x^2 + 10x + 25.

Square with side x + 5 split into an x by x square (x squared), two x by 5 rectangles (5x each) and a 5 by 5 square (25). Area = (x + 5) squared = x squared + 10x + 25.x²5x5x25x5x5Side: x + 5Area: (x + 5)²= x² + 5x + 5x + 25= x² + 10x + 25

Box (rectangular prism). V = \text{length} \times \text{width} \times \text{height}. If all three contain x, you get an x^3 term.

Uniform borders. A border of width x goes on both ends of each side. So a side of 8 becomes 8 + 2x, not 8 + x. In the figure below, an 8-inch by 10-inch photo has a frame of width x:

\text{Outer area} = (8 + 2x)(10 + 2x) = 80 + 16x + 20x + 4x^2 = 4x^2 + 36x + 80 \text{Frame only} = \text{outer area} - \text{photo} = (4x^2 + 36x + 80) - 80 = 4x^2 + 36x

Cutting corners from a sheet. If you cut an x-by-x square from each corner of a sheet and fold up the sides, each side of the base loses 2x (one x from each end), and the height of the box is x.

An 8 inch by 10 inch photo with a frame border of width x on all four sides. The outer width is 8 + 2x inches and the outer height is 10 + 2x inches.photo8 in × 10 in8 in10 inxxxx8 + 2x10 + 2x

A border of width x adds 2x to each dimension: the outer frame is (8 + 2x) by (10 + 2x) inches.

Revenue and "for each" changes

Business questions often say something like: "A theater sells 300 tickets at $12. For each $1 increase in price, it sells 20 fewer tickets."

Build two linear pieces, then multiply them.

  1. Let x = the number of $1 increases.
  2. Price = 12 + x dollars.
  3. Tickets sold = 300 - 20x (20 fewer for each increase, so the 20 is multiplied by x).
  4. Revenue = price × tickets = (12 + x)(300 - 20x).
Revenue as the area of a rectangle: height = price, width = tickets sold. With x = 0 the rectangle is 12 dollars by 300 tickets, area 3600 dollars. With x = 1 it is 13 dollars by 280 tickets, area 3640 dollars. In general revenue = (12 + x)(300 − 20x).$3600300 tickets$12x = 0$3640300 − 20(1) = 28012 + 1 = $13x = 1Revenue = price × tickets = (12 + x)(300 − 20x)

Multiply out with FOIL:

(12 + x)(300 - 20x) = 3600 - 240x + 300x - 20x^2 = -20x^2 + 60x + 3600

Check with a number. If x = 1: price $13, tickets 280, revenue 13 \cdot 280 = 3640. The polynomial gives -20 + 60 + 3600 = 3640 ✓.

Consecutive-number problems work the same way. "The product of two consecutive even integers, where n is the smaller one" is n(n + 2) = n^2 + 2n.

A good habit for every model: after writing the expression, plug in a small number like x = 1 or x = 2 and compare it with what the story says should happen. If the numbers match, your expression is almost certainly right.

Formula

Formulas you will use

A_{\text{rectangle}} = l \cdot w \qquad P_{\text{rectangle}} = 2l + 2w \qquad A_{\text{square}} = s^2 V_{\text{box}} = l \cdot w \cdot h \quad (\text{on the sheet: } V = lwh \text{ or } V = Bh) \text{Revenue (total cost)} = (\text{price per unit}) \times (\text{number of units}) (a + b)^2 = a^2 + 2ab + b^2 \qquad (a + b)(c + d) = ac + ad + bc + bd
  • l = length, w = width, h = height, s = side of a square, B = area of the base.

On the GED formula sheet: rectangle area A = lw and perimeter P = 2l + 2w, square area A = s^2, rectangular-prism volume V = lwh (also V = Bh), and "total cost = (number of units) × (price per unit)."

Memorize: the FOIL / square-of-a-sum patterns — they are not printed on the sheet.

Worked example

Example 1 (easy): Two phrases that look alike

Write and simplify an expression for each phrase. Let n be the number.

Example 1 with n = 8, drawn as unit squares. (a) Five less than the square of n: an 8 by 8 grid of 64 squares with 5 squares crossed out, leaving 59. (b) The square of five less than n: only a 3 by 3 square (side n − 5 = 3) is shaded, giving 9. The two phrases give very different values.n = 8(a) n² − 564 − 5 = 59n − 5 = 3(b) (n − 5)²3² = 9

(a) "Five less than the square of a number." (b) "The square of five less than a number."

  1. (a) "The square of a number" is n^2. "Five less than" that means subtract 5 from it: n^2 - 5.
  2. (b) "Five less than a number" is n - 5. Now square the whole group: (n - 5)^2.
  3. Expand (b): (n - 5)(n - 5) = n^2 - 5n - 5n + 25 = n^2 - 10n + 25.

Answer: (a) n^2 - 5; (b) n^2 - 10n + 25

Check: Let n = 8. (a) 64 - 5 = 59 ✓. (b) 8 - 5 = 3, and 3^2 = 9; the polynomial gives 64 - 80 + 25 = 9 ✓.

Worked example

Example 2 (work backwards): A rectangle's missing side

The length of a rectangle is 3 feet more than twice its width, w. (a) Write the area as a polynomial. (b) The area of a second rectangle is 2w^2 + 3w square feet and its width is w. What is its length?

Example 2, drawn to scale for w = 4: a rectangle of width w and length 2w + 3, split into a w by 2w part (area 2w squared) and a w by 3 part (area 3w). Area = w(2w + 3) = 2w squared + 3w; with w = 4 that is 4 by 11 = 44 square feet.w · 2w = 2w²3w2w3wlength = 2w + 3Area = w(2w + 3) = 2w² + 3wDrawn for w = 4: 4 × 11 = 44 sq ft
  1. (a) Length = 2w + 3. Area = w(2w + 3).
  2. Distribute: w \cdot 2w + w \cdot 3 = 2w^2 + 3w.
  3. (b) Work backwards: area ÷ width = length. Factor out w: 2w^2 + 3w = w(2w + 3), so the length is 2w + 3.

Answer: (a) 2w^2 + 3w square feet; (b) 2w + 3 feet

Check: Let w = 4. Length = 11, area = 44. The polynomial gives 2(16) + 12 = 44 ✓.

Worked example

Example 3 (real world): A picture frame

An 8-inch by 10-inch photo gets a frame of uniform width x inches (see the figure above). Write a simplified expression for the area of the frame only.

Example 3 area model. The outer frame is (8 + 2x) by (10 + 2x). Rearranged, it splits into the 8 by 10 photo (80) plus three shaded frame pieces: 2x by 8 is 16x, 2x by 10 is 20x and 2x by 2x is 4x squared. Frame only = 16x + 20x + 4x squared = 4x squared + 36x square inches.photo 8016x20x4x²102x82x10 + 2x8 + 2xOuter area:(8 + 2x)(10 + 2x)= 80 + 16x + 20x + 4x²Frame only (shaded):16x + 20x + 4x²= 4x² + 36x
  1. Outer width = 8 + 2x; outer height = 10 + 2x (the border is on both sides).
  2. Outer area = (8 + 2x)(10 + 2x).
  3. FOIL: 80 + 16x + 20x + 4x^2 = 4x^2 + 36x + 80.
  4. Subtract the photo's area, 8 \cdot 10 = 80: 4x^2 + 36x + 80 - 80 = 4x^2 + 36x.

Answer: 4x^2 + 36x square inches

Check: Let x = 1. Outer: 10 \cdot 12 = 120; frame = 120 - 80 = 40. Polynomial: 4 + 36 = 40 ✓.

Worked example

Example 4 (real world): Ticket revenue

A theater sells 300 tickets at $12 each. For every $1 increase in price, it sells 20 fewer tickets. Let x be the number of $1 increases. Write the revenue as a simplified polynomial.

Example 4 table and graph of revenue R = −20x squared + 60x + 3600. Table: x = 0, price 12 dollars, 300 tickets, 3600 dollars; x = 1, 13 dollars, 280, 3640 dollars; x = 2, 14 dollars, 260, 3640 dollars; x = 3, 15 dollars, 240, 3600 dollars (highlighted, the check); x = 4, 16 dollars, 220, 3520 dollars; x = 5, 17 dollars, 200, 3400 dollars; x = 6, 18 dollars, 180, 3240 dollars. The graph plots these points; revenue rises slightly then falls.xpriceticketsrevenue0$12300$36001$13280$36402$14260$36403$15240$36004$16220$35205$17200$34006$18180$32403200330034003500360037000123456number of $1 increases, xrevenue ($)(3, 3600)
  1. Price: 12 + x dollars.
  2. Tickets: 300 - 20x.
  3. Revenue: (12 + x)(300 - 20x).
  4. FOIL: 12 \cdot 300 = 3600; 12 \cdot (-20x) = -240x; x \cdot 300 = 300x; x \cdot (-20x) = -20x^2.
  5. Combine: -20x^2 + 60x + 3600.

Answer: -20x^2 + 60x + 3600 dollars

Check: x = 3: price $15, tickets 240, revenue 15 \cdot 240 = 3600. Polynomial: -180 + 180 + 3600 = 3600 ✓.

Worked example

Example 5 (GED-hard): An open box

A 20-inch by 12-inch piece of cardboard has an x-inch square cut from each corner. The sides are folded up to make an open box. Write the volume as a simplified polynomial.

Example 5, drawn to scale with x = 2: a 20-inch by 12-inch sheet with an x by x square cut from each corner (red) and dashed fold lines. The base left in the middle is (20 − 2x) by (12 − 2x). Folded up, the open box has length 20 − 2x, width 12 − 2x and height x, so V = x(20 − 2x)(12 − 2x) = 4x cubed − 64x squared + 240x.20 in12 inxbase(20 − 2x) by (12 − 2x)20 − 2x12 − 2xxopen boxV = x(20 − 2x)(12 − 2x) = 4x³ − 64x² + 240x
  1. Each side of the base loses x at both ends: length = 20 - 2x, width = 12 - 2x.
  2. The folded flaps are the height: h = x.
  3. Volume = x(20 - 2x)(12 - 2x).
  4. Multiply the two binomials first: (20 - 2x)(12 - 2x) = 240 - 40x - 24x + 4x^2 = 4x^2 - 64x + 240.
  5. Multiply by x: 4x^3 - 64x^2 + 240x.

Answer: V = 4x^3 - 64x^2 + 240x cubic inches

Check: x = 2: the box is 16 \times 8 \times 2 = 256. Polynomial: 4(8) - 64(4) + 240(2) = 32 - 256 + 480 = 256 ✓.

Calculator tip

Calculator tip: test your model with a number

The TI-30XS MultiView cannot multiply polynomials for you, but it is perfect for checking that a model and a simplified form agree.

  1. Choose a simple test value, such as x = 2.
  2. Compute the story directly. For the box in Example 5: 2 × 16 × 8 enter → 256.
  3. Compute an answer choice. For 4x^3 - 64x^2 + 240x: 4 × 2 ^ 3 ▶ − 64 × 2 x² + 240 × 2 enter → 256 ✓.
  4. Use the (−) key for negative values and put them in parentheses before squaring or cubing.

When not to use it: writing the model (choosing what to multiply) is reading, not arithmetic. Do that on your erasable note board first. The calculator only confirms.

Common mistake

Common mistakes

  • Border on one side only. Wrong: a 20-ft pool with an x-ft walkway is 20 + x wide. Right: 20 + 2x, because the walkway is on both sides.
  • Squaring only part of a group. Wrong: (x + 3)^2 = x^2 + 9. Right: x^2 + 6x + 9. The middle term 2 \cdot 3 \cdot x is easy to forget.
  • "Less than" order. Wrong: "7 less than n^2" = 7 - n^2. Right: n^2 - 7.
  • Forgetting "for each." Wrong: tickets = 300 - 20. Right: 300 - 20x, because 20 fewer are sold for each $1 increase.
  • Using the wrong formula. Wrong: writing the perimeter 2w + 2(w + 4) when the question asks for area. Right: area = w(w + 4).
  • Forgetting the height. Wrong: open-box volume = (20 - 2x)(12 - 2x). Right: multiply by the height x as well.
Test strategy

GED test strategy

  • Underline the quantities. Before writing anything, underline every number and every "for each," "more than," "less than," "times."
  • Draw it. For shapes, a quick sketch with labels like 20 + 2x prevents the one-side-border mistake.
  • Plug in a number to test choices. Pick x = 1 or x = 2, work out the real-world answer directly, then evaluate each choice. Wrong choices almost never match.
  • Check the degree. Area of a shape with two variable sides has an x^2 term; volume with three variable dimensions has an x^3 term; perimeter has neither.
  • Read the question twice: total area or border only? Revenue or number sold? Simplified or unsimplified form?
  • Time: these questions reward 20 seconds of careful reading more than fast algebra.

Check yourself

Write polynomial expressions from contextNo calculator

Which expression represents "the square of a number n, decreased by 7"?

Answer choices
Hint

Find "the square of a number" first. Then decide what happens to that result.

Check yourself

Write polynomial expressions from contextNo calculator

The length of a rectangular rug is 4 feet more than its width, w. Which expression represents the area of the rug, in square feet?

Answer choices
Hint

Write the length in terms of w. Area = length × width.

Summary

  • Translate one phrase at a time; "less than" flips the order and "the square of" applies to the whole group that follows.
  • Put variable sides in parentheses: area = l \cdot w, volume = l \cdot w \cdot h.
  • A uniform border of width x adds 2x to each dimension; cut corners remove 2x and leave height x.
  • Revenue = (price) × (number sold), where a "for each" change multiplies the variable.
  • Multiply out with FOIL to get simplified form, then check with a test number.
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