UNIT 1PolynomialsLesson 1 of 31 Adding, Subtracting and Multiplying Polynomials2 Special Products3 Factoring: GCF and Grouping
Adding, Subtracting and Multiplying Polynomials
A polynomial is a sum of terms like 3x² and −5x. You can add, subtract and multiply polynomials.
What you'll be able to do
- Name the degree, the leading coefficient and the type.
- Add polynomials, and subtract by changing every sign.
- Multiply by a monomial, a binomial or a trinomial.
- Write areas and volumes as polynomials.
The big idea
Like terms. Terms are alike when they have the same letters with the same exponents. 3x² and 5x² are alike. 3x² and 5x are not.
Standard form. Write the terms from the highest exponent to the lowest: 4x³ − 2x² + 7.
Add like terms. (2x² + 3x) + (x² − 5x) = 3x² − 2x.
Subtract by changing every sign. −(x² − 4x + 1) = −x² + 4x − 1. Then add.
Add exponents when you multiply. x² · x³ = x⁵.
Multiply every term by every term. In (x + 2)(x + 5), multiply x and 2 by both x and 5. You get x² + 5x + 2x + 10 = x² + 7x + 10.
Count the products. A binomial times a trinomial gives 2 × 3 = 6 products before you add like terms.
Words you need
- Polynomial
- A sum of terms. Each term is a number times letters with whole-number exponents.
- Term
- One part of a polynomial, like 3x² or −5.
- Monomial
- A polynomial with 1 term, like 4x³.
- Binomial
- A polynomial with 2 terms, like x + 5.
- Trinomial
- A polynomial with 3 terms, like x² + 7x + 10.
- Like terms
- Terms with the same letters and the same exponents, like 3x² and 5x².
- Degree
- The highest exponent in a polynomial. x³ + 2x has degree 3.
- Coefficient
- The number in front of a letter. In −5x², it is −5.
- Leading coefficient
- The coefficient of the term with the highest exponent.
- Constant
- A term with no letter, like 7.
- Standard form
- The terms written from the highest exponent to the lowest.
- Distributive property
- a(b + c) = ab + ac. Multiply the outside by each term inside.
- Area model
- A grid that shows each product as a box. The boxes add up to the answer.
- Base
- The bottom of a shape, like the bottom of a box. In a power like 3², the base is 3.
- Exponent
- The small raised number. In x³, it is 3.
- Power
- A base with an exponent, like x³.
- Variable
- A letter that stands for a number, like x.
- Expression
- Numbers and letters joined by +, −, × or ÷. It has no = sign.
- Simplify
- Write it in a shorter form with the same value.
- Product
- The answer when you multiply.
- Factor
- A part you multiply. In x(x − 2)(x − 5), each part is a factor.
- Zero
- An x-value that makes a function equal 0.
Area model for (3x + 2)(x + 5)
| Term | Times the first term | Times the second term |
|---|---|---|
| 3x | 3x² | 15x |
| 2 | 2x | 10 |
Worked examples
Example 1
Write 5x − 2x³ + 8 − x in standard form. Give its degree, its leading coefficient and its type.
- Add the like terms: 5x − x = 4x.
- Order the terms from the highest exponent: −2x³ + 4x + 8.
- The highest exponent is 3, so the degree is 3. Its coefficient, −2, is the leading coefficient.
- It has 3 terms, so it is a trinomial.
Answer: −2x³ + 4x + 8. Degree 3, leading coefficient −2, trinomial
Example 2
Subtract (4x² − 3x + 2) − (x² − 5x + 7).
- Change every sign in the second polynomial: −x² + 5x − 7.
- Now add: 4x² − 3x + 2 − x² + 5x − 7.
- Add the like terms: 3x² + 2x − 5.
Answer: 3x² + 2x − 5
Example 3
Multiply (3x + 2)(x + 5). Use the area model above.
- Each box is a product: 3x · x = 3x², 3x · 5 = 15x, 2 · x = 2x and 2 · 5 = 10.
- Add the four boxes: 3x² + 15x + 2x + 10.
- Add the like terms 15x and 2x: 3x² + 17x + 10.
Answer: 3x² + 17x + 10
Example 4
Multiply (x − 2)(x² + 3x − 4).
- Multiply x by each term: x³ + 3x² − 4x.
- Multiply −2 by each term: −2x² − 6x + 8.
- Add the like terms: x³ + x² − 10x + 8.
Answer: x³ + x² − 10x + 8
Common mistakes
Changing only the first sign. In (5x² + x) − (2x² − 3x), the −3x becomes +3x. The answer is 3x² + 4x.
Multiplying exponents. x² · x⁴ = x⁶, not x⁸.
Skipping the middle terms. (x + 5)(x + 2) is not x² + 10. It is x² + 7x + 10.
Adding unlike terms. 3x² + 2x is not 5x³. The terms are not alike, so it stays 3x² + 2x.
Practice: 20 questions
12 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Write each polynomial in standard form, state its degree and leading coefficient, and classify it as a monomial, binomial, or trinomial: Polynomial A is 7x − 4x³ + 9, and Polynomial B is 5y² − 8y⁵.
- Aoife fences a triangular dog run whose three sides measure 3x + 4 feet, 2x² − x + 5 feet, and x² + 6x − 2 feet. Write a polynomial in standard form for the perimeter of the dog run in feet, and find the perimeter when x = 4. Word problem
- Tiago subtracts (5x² + 3x − 8) − (2x² − 4x + 6) by writing 5x² + 3x − 8 − 2x² − 4x + 6 and simplifying to 3x² − x − 2. Find his mistake and write the correct polynomial in standard form.
- Chisom designs two rectangular banners. Banner A has a width of 3x inches and a length of 2x + 5 inches. Banner B has a width of 2x inches and a length of 3x + 7 inches. Write a polynomial in standard form for the area of each banner in square inches. When x is a positive number, which banner has the greater area, and by how many square inches in terms of x? Word problem
- Dmitri builds a rectangular vegetable garden that is 2x + 3 meters long and x + 4 meters wide. Write a polynomial in standard form for the area of the garden in square meters. Word problem
Tier 2 Intermediate One more step to think through
- Sunita simplifies two expressions on her worksheet. First, she writes 2x²(3x³ − 4) = 6x⁶ − 8x². Second, she writes (x + 3)(x + 4) = x² + 12. Find her mistake in each expression and write the correct simplified polynomial in standard form for each one.
- Gael runs a custom skateboard shop. Selling x skateboards brings in a revenue of −2x² + 120x dollars. Building x skateboards costs 3x² − 15x + 400 dollars. Profit equals revenue minus cost. Write a polynomial in standard form for Gael's profit in dollars from x skateboards, and find his profit when he sells 10 skateboards. Word problem
- In the equation (3x + k)(2x − 5) = 6x² − 7x − 20, find the number k.
- Ayumi starts with a square patio that measures x feet on each side, where x is greater than 5. Design 1 increases the length by 6 feet and decreases the width by 2 feet. Design 2 increases the length by 5 feet and decreases the width by 1 foot. Write a polynomial in standard form for the area of each design in square feet. Which design has the greater area for every x greater than 5, and by how many square feet? Word problem
- Femi packs glassware into a shipping box whose rectangular base has an area of x² + 3x + 2 square inches and whose height is 2x + 5 inches. Write a polynomial in standard form for the volume of the box in cubic inches. Word problem
Tier 3 Advanced Several steps - find the hidden structure
- Zosia builds a brick walkway of uniform width x meters all the way around the outside of a rectangular pool that measures 10 meters wide by 16 meters long. Write a polynomial in standard form for the total area of the pool and walkway. Then write a polynomial in standard form for the area of the brick walkway alone, in square meters. Word problem
- Emeka starts with a flat rectangular sheet of cardboard that is 14 inches wide and 20 inches long. He cuts a square of side length x inches out of each of the four corners and folds up the sides to make an open-top box. Write a polynomial in standard form for the volume of the box in cubic inches, and state its degree and leading coefficient. Word problem
- Vahan simplifies 5x(2x − 3) − (x − 4)(3x + 2). He correctly expands (x − 4)(3x + 2) to 3x² − 10x − 8, then writes 10x² − 15x − 3x² − 10x − 8 and simplifies to 7x² − 25x − 8. Find his mistake and write the correct polynomial in standard form.
- Noor compares the volumes of two wooden crates. Crate A has dimensions x feet by x + 2 feet by x + 4 feet. Crate B has a square base of side x + 2 feet and a height of x feet. Write each crate's volume in cubic feet as a polynomial in standard form. For every positive number x, which crate has the greater volume, and by how many cubic feet in terms of x? Word problem
- When (2x − 3)(x² + kx + 4) is multiplied out and written in standard form, the coefficient of the x² term is 7. Find the number k, and write the complete expanded polynomial in standard form.
Tier 4 Challenge Stretch problems, some above grade level
- Cian lays two identical rectangular rugs of length 2x + 5 feet and width x + 3 feet on a living room floor. The two rugs overlap in a rectangular region that measures x + 1 feet by x − 2 feet, where x is greater than 2. Write a polynomial in standard form for the total floor area covered by the two rugs in square feet. Word problem
- Lulit looks at the polynomials P = 4x³ − 5x² + 2x − 1 and Q = 4x³ + 3x² − 7x + 4. She claims that because P and Q both have degree 3, the difference P − Q must also have degree 3. Find her mistake, simplify P − Q in standard form, and state its actual degree and leading coefficient.
- Stellan has two identical rectangular courtyards that each measure 3x meters long by 2x meters wide. In Courtyard A, he adds a 2-meter-wide paved strip along one length and one width, expanding the courtyard to 3x + 2 meters by 2x + 2 meters. In Courtyard B, he increases the length by 4 meters to 3x + 4 meters and keeps the width at 2x meters. Find the polynomial in standard form for the new area added to Courtyard A and for the new area added to Courtyard B in square meters. For which positive values of x does Courtyard A gain more area than Courtyard B, and by how many square meters in terms of x? Word problem
- Multiply (x + 3)(x² + bx + c). Find the numbers b and c so that when the product is written in standard form, both the x² term and the x term have a coefficient of 0, leaving a binomial. Then write that resulting binomial.
- A zero of a polynomial function is an x-value that makes the output y equal 0. Indira models a new company's monthly profit y, in thousands of dollars, at x months after launch with the polynomial function y = x(x − 2)(x − 5) for x ≥ 0. First, find all three zeroes of this function and state what a zero means for the company's profit. Next, multiply out x(x − 2)(x − 5) to write the polynomial in standard form, and find the company's profit or loss at x = 1 month and at x = 3 months. Word problem Above grade level
Answers
Tier 1 Foundations
- Polynomial A: −4x³ + 7x + 9, degree 3, leading coefficient −4, trinomial; Polynomial B: −8y⁵ + 5y², degree 5, leading coefficient −8, binomial. Standard form orders terms from highest exponent to lowest exponent: Polynomial A becomes −4x³ + 7x + 9, and Polynomial B becomes −8y⁵ + 5y². For −4x³ + 7x + 9, the highest exponent is 3, the coefficient of that term is −4, and it has 3 terms, making it a trinomial. For −8y⁵ + 5y², the highest exponent is 5, the coefficient of that term is −8, and it has 2 terms, making it a binomial.
- 3x² + 8x + 7 feet; 87 feet when x = 4. Add the three side lengths and group like terms: (2x² + x²) + (3x − x + 6x) + (4 + 5 − 2). Combine like terms in standard form: 3x² + 8x + 7 feet. Substitute x = 4: 3(4)² + 8(4) + 7 = 48 + 32 + 7 = 87 feet.
- Tiago subtracted only the first term 2x² instead of changing the sign of every term in the second polynomial; the correct answer is 3x² + 7x − 14. Distribute the subtraction sign across every term in the second set of parentheses: −(2x² − 4x + 6) = −2x² + 4x − 6. Group like terms: (5x² − 2x²) + (3x + 4x) + (−8 − 6). Combine like terms: 3x² + 7x − 14.
- Banner A has area 6x² + 15x square inches and Banner B has area 6x² + 14x square inches; Banner A is greater by x square inches. Multiply width by length for Banner A: 3x(2x + 5) = 6x² + 15x square inches. Multiply width by length for Banner B: 2x(3x + 7) = 6x² + 14x square inches. Subtract the areas: (6x² + 15x) − (6x² + 14x) = x square inches. Since x > 0, Banner A is always greater by x square inches.
- 2x² + 11x + 12 square meters. Multiply the length and width: (2x + 3)(x + 4). Distribute each term in the first binomial across the second binomial: 2x(x + 4) + 3(x + 4) = 2x² + 8x + 3x + 12. Combine the middle like terms: 2x² + 11x + 12 square meters.
Tier 2 Intermediate
- In the first expression she multiplied the exponents 2 and 3 instead of adding them, and the correct answer is 6x⁵ − 8x²; in the second expression she forgot to multiply the outer and inner terms 4x and 3x, and the correct answer is x² + 7x + 12. When multiplying x² · x³, add the exponents to get x⁵, so 2x²(3x³ − 4) = 6x⁵ − 8x². When multiplying (x + 3)(x + 4), distribute every term: x(x + 4) + 3(x + 4) = x² + 4x + 3x + 12. Combine like terms in the second expression to get x² + 7x + 12.
- −5x² + 135x − 400 dollars; $450 when x = 10. Set up revenue minus cost: (−2x² + 120x) − (3x² − 15x + 400). Change every sign in the cost polynomial: −2x² + 120x − 3x² + 15x − 400. Combine like terms to get −5x² + 135x − 400. At x = 10, profit is −5(100) + 135(10) − 400 = −500 + 1,350 − 400 = $450.
- k = 4. Expand the left side by distributing: (3x + k)(2x − 5) = 6x² − 15x + 2kx − 5k = 6x² + (2k − 15)x − 5k. Match the constant terms: −5k = −20, which gives k = 4. Check the x-coefficient with k = 4: 2(4) − 15 = 8 − 15 = −7, which matches −7x.
- Design 1 has area x² + 4x − 12 square feet and Design 2 has area x² + 4x − 5 square feet; Design 2 has the greater area by 7 square feet. Expand the area of Design 1: (x + 6)(x − 2) = x² − 2x + 6x − 12 = x² + 4x − 12 square feet. Expand the area of Design 2: (x + 5)(x − 1) = x² − x + 5x − 5 = x² + 4x − 5 square feet. Subtract Design 1 from Design 2: (x² + 4x − 5) − (x² + 4x − 12) = 7 square feet.
- 2x³ + 11x² + 19x + 10 cubic inches. Multiply the height by the base area: (2x + 5)(x² + 3x + 2). Distribute 2x and 5 across the trinomial: 2x(x² + 3x + 2) + 5(x² + 3x + 2) = 2x³ + 6x² + 4x + 5x² + 15x + 10. Combine like terms: 2x³ + 11x² + 19x + 10 cubic inches.
Tier 3 Advanced
- Total combined area is 4x² + 52x + 160 square meters; walkway area alone is 4x² + 52x square meters. Because the walkway of width x is on both ends of the width and both ends of the length, the outer rectangle measures 2x + 10 meters wide by 2x + 16 meters long. Multiply the outer dimensions: (2x + 10)(2x + 16) = 4x² + 32x + 20x + 160 = 4x² + 52x + 160 square meters. Subtract the pool's area of 10 × 16 = 160 square meters to get the walkway area alone: 4x² + 52x square meters.
- 4x³ − 68x² + 280x cubic inches; degree 3; leading coefficient 4. Cutting a square of side x from both ends of each side leaves a base width of 14 − 2x inches, a base length of 20 − 2x inches, and a folded height of x inches. Multiply the base dimensions: (14 − 2x)(20 − 2x) = 280 − 28x − 40x + 4x² = 4x² − 68x + 280. Multiply the base area by the height x and write in standard form: x(4x² − 68x + 280) = 4x³ − 68x² + 280x cubic inches, which has degree 3 and leading coefficient 4.
- Vahan did not keep 3x² − 10x − 8 in parentheses when subtracting, so he failed to flip the signs of −10x and −8; the correct polynomial is 7x² − 5x + 8. Expand each product first: 5x(2x − 3) = 10x² − 15x and (x − 4)(3x + 2) = 3x² − 10x − 8. Subtract the entire trinomial by changing every sign inside: (10x² − 15x) − (3x² − 10x − 8) = 10x² − 15x − 3x² + 10x + 8. Combine like terms: 7x² − 5x + 8.
- Crate A has volume x³ + 6x² + 8x cubic feet and Crate B has volume x³ + 4x² + 4x cubic feet; Crate A has the greater volume by 2x² + 4x cubic feet. For Crate A, multiply (x + 2)(x + 4) = x² + 6x + 8, then multiply by x to get x³ + 6x² + 8x cubic feet. For Crate B, multiply (x + 2)(x + 2) = x² + 4x + 4, then multiply by x to get x³ + 4x² + 4x cubic feet. Subtract Crate B from Crate A: (x³ + 6x² + 8x) − (x³ + 4x² + 4x) = 2x² + 4x cubic feet, which is positive for every x > 0.
- k = 5; 2x³ + 7x² − 7x − 12. Distribute to find the x² terms: 2x(kx) = 2kx² and −3(x²) = −3x², so the combined x² term is (2k − 3)x². Set 2k − 3 = 7, add 3 to get 2k = 10, and divide by 2 to find k = 5. Expand (2x − 3)(x² + 5x + 4) = 2x³ + 10x² + 8x − 3x² − 15x − 12 = 2x³ + 7x² − 7x − 12.
Tier 4 Challenge
- 3x² + 23x + 32 square feet. Find the area of one rug: (2x + 5)(x + 3) = 2x² + 11x + 15 square feet, so two rugs have a combined area of 2(2x² + 11x + 15) = 4x² + 22x + 30 square feet. Find the area of the overlapping region: (x + 1)(x − 2) = x² − x − 2 square feet. Because the overlap is counted twice in the sum of the two rugs, subtract it once: (4x² + 22x + 30) − (x² − x − 2) = 3x² + 23x + 32 square feet.
- The 4x³ terms cancel out when subtracting, so the degree drops below 3; P − Q = −8x² + 9x − 5, which has degree 2 and leading coefficient −8. Set up the subtraction: (4x³ − 5x² + 2x − 1) − (4x³ + 3x² − 7x + 4). Distribute the negative sign: 4x³ − 5x² + 2x − 1 − 4x³ − 3x² + 7x − 4. Combine like terms: 4x³ − 4x³ = 0, leaving −8x² + 9x − 5, which is a trinomial of degree 2 with leading coefficient −8.
- Courtyard A gains 10x + 4 square meters and Courtyard B gains 8x square meters; Courtyard A gains more area for every positive value of x, by 2x + 4 square meters. Each courtyard starts with area 3x · 2x = 6x² square meters. Courtyard A's expanded area is (3x + 2)(2x + 2) = 6x² + 10x + 4, so the added area is (6x² + 10x + 4) − 6x² = 10x + 4 square meters. Courtyard B's expanded area is 2x(3x + 4) = 6x² + 8x, so the added area is 8x square meters. Subtracting gives (10x + 4) − 8x = 2x + 4, which is positive for every x > 0.
- b = −3 and c = 9; the resulting binomial is x³ + 27. Distribute (x + 3)(x² + bx + c) = x³ + bx² + cx + 3x² + 3bx + 3c = x³ + (b + 3)x² + (c + 3b)x + 3c. Set the x² coefficient equal to 0: b + 3 = 0, which gives b = −3. Set the x coefficient equal to 0 using b = −3: c + 3(−3) = 0, which gives c = 9. The constant term is 3(9) = 27, giving x³ + 27.
- The zeroes are x = 0, x = 2, and x = 5, which are the months when the company breaks even with $0 profit; the standard form is y = x³ − 7x² + 10x; at x = 1 the company makes a $4,000 profit, and at x = 3 it has a $6,000 loss. A product equals 0 when any factor is 0, giving x = 0, x − 2 = 0 (so x = 2), and x − 5 = 0 (so x = 5). Multiply the binomials first: (x − 2)(x − 5) = x² − 7x + 10. Then multiply by x to get y = x³ − 7x² + 10x. At x = 1, y = 1 − 7 + 10 = 4, which is $4,000 profit; at x = 3, y = 27 − 63 + 30 = −6, which is a $6,000 loss.
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