UNIT 1PolynomialsLesson 2 of 31 Adding, Subtracting and Multiplying Polynomials2 Special Products3 Factoring: GCF and Grouping
Special Products
Three products come up again and again. Learn their patterns, and you can multiply them in one step.
What you'll be able to do
- Expand (a + b)² and (a − b)² in one step.
- Expand (a + b)(a − b) as a² − b².
- Tell if a trinomial is a perfect square.
- Use the patterns for mental math and areas.
The big idea
Square of a sum. (a + b)² = a² + 2ab + b². So (x + 3)² = x² + 6x + 9.
Square of a difference. (a − b)² = a² − 2ab + b². The last term is always plus: (x − 3)² = x² − 6x + 9.
Sum times difference. (a + b)(a − b) = a² − b². The middle terms cancel: (x + 3)(x − 3) = x² − 9.
The middle term is 2ab. In the area model, it is the two rectangles. So (x + 3)² is not x² + 9.
Perfect square test. A trinomial is a perfect square when its first and last terms are squares and its middle term is 2ab. 4x² + 12x + 9 = (2x + 3)².
Mental math. 21 × 19 = (20 + 1)(20 − 1) = 400 − 1 = 399. And 31² = 900 + 60 + 1 = 961.
Words you need
- Binomial
- A polynomial with 2 terms, like x + 3.
- Trinomial
- A polynomial with 3 terms, like x² + 6x + 9.
- Term
- One part of a polynomial, like 6x or 9.
- Expand
- Multiply out the brackets and write the result as a polynomial.
- Square of a sum
- (a + b)² = a² + 2ab + b².
- Square of a difference
- (a − b)² = a² − 2ab + b².
- Difference of squares
- a² − b². It equals (a + b)(a − b).
- Perfect square trinomial
- A trinomial that equals a binomial squared, like x² + 6x + 9 = (x + 3)².
- Area model
- A grid that shows each product as a box. The boxes add up to the answer.
- Coefficient
- The number in front of a letter. In 6x, it is 6.
- Constant
- A term with no letter, like 9.
- Exponent
- The small raised number. In x², it is 2.
- Power
- A base with an exponent, like 3².
- Product
- The answer when you multiply.
- 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.
- Factor
- A part you multiply. In (x + 3)(x − 3), each binomial is a factor.
- Quadratic
- A polynomial whose highest exponent is 2, like x² − 12x + 41.
- Zero
- An x-value that makes a function equal 0.
The three patterns
| Product | Pattern | Example |
|---|---|---|
| (a + b)² | a² + 2ab + b² | (x + 4)² = x² + 8x + 16 |
| (a − b)² | a² − 2ab + b² | (x − 7)² = x² − 14x + 49 |
| (a + b)(a − b) | a² − b² | (x + 6)(x − 6) = x² − 36 |
Worked examples
Example 1
Expand (2x − 5)².
- Use (a − b)² = a² − 2ab + b² with a = 2x and b = 5.
- a² = (2x)² = 4x². 2ab = 2(2x)(5) = 20x. b² = 25.
- So (2x − 5)² = 4x² − 20x + 25.
Answer: 4x² − 20x + 25
Example 2
Expand (3m + 4n)(3m − 4n).
- This is a sum times a difference, so it is a² − b² with a = 3m and b = 4n.
- a² = 9m² and b² = 16n².
- So the product is 9m² − 16n². The middle terms 12mn and −12mn cancel.
Answer: 9m² − 16n²
Example 3
Find 52 × 48 and 101² without a calculator.
- 52 × 48 = (50 + 2)(50 − 2) = 50² − 2² = 2,500 − 4 = 2,496.
- 101² = (100 + 1)² = 100² + 2(100)(1) + 1².
- That is 10,000 + 200 + 1 = 10,201.
Answer: 52 × 48 = 2,496 and 101² = 10,201
Example 4
Is 9x² − 24x + 16 a perfect square trinomial? Is 9x² − 12x + 16?
- 9x² = (3x)² and 16 = 4². For a perfect square, the middle term must be 2(3x)(4) = 24x, with a minus sign here.
- 9x² − 24x + 16 has −24x. So it is (3x − 4)².
- 9x² − 12x + 16 has −12x, not −24x. So it is not a perfect square.
Answer: 9x² − 24x + 16 = (3x − 4)² is a perfect square. 9x² − 12x + 16 is not
Common mistakes
Dropping the middle term. (x + 8)² is x² + 16x + 64, not x² + 64.
A minus sign on the last term. (x − 9)² = x² − 18x + 81. The last term is +81, because (−9)² = 81.
Squaring only the letter. (3x)² is 9x², not 3x².
Mixing up the patterns. (x − 2)² is not x² − 4. That is (x + 2)(x − 2).
Practice: 20 questions
11 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Use special-product patterns to expand (x + 6)², (y − 4)², and (m + 9)(m − 9). Write each answer in standard form.
- Callum builds a square tabletop whose side length is x + 5 inches. To find the area of the tabletop, he writes (x + 5)² = x² + 25 square inches. Explain why his expansion is missing area when a square of side x + 5 is split into four rectangular sections, and write the correct polynomial in standard form. Word problem
- Zuleika has two separate square gardens, one with side length a meters and one with side length b meters, where a and b are positive numbers. Her neighbor has one single square garden whose side length is a + b meters. Write an expression for the combined area of Zuleika's two gardens and an expanded expression for her neighbor's garden area in square meters. Whose total garden area is larger for all positive values of a and b, and by how many square meters? Word problem
- Find the missing positive numbers that complete these two special-product equations: (x + p)² = x² + 14x + q and (y − r)² = y² − sy + 36.
- Kenzo orders 49 crates of apples for a market, and each crate holds 51 apples. Find the exact total number of apples without a calculator or long multiplication. Word problem
Tier 2 Intermediate One more step to think through
- Expand (3x − 2y)² and (4a + 5b)(4a − 5b). Write each simplified polynomial.
- Daria completes two problems on her quiz. First, she expands (2x − 3)² as 4x² − 12x − 9. Second, she claims that x² + 10x + 100 is a perfect-square trinomial because x² and 100 are both perfect squares. Find her mistake in each problem, write the correct expansion of (2x − 3)², and state what the middle term of x² + __ + 100 would need to be for it to equal (x + 10)².
- Obi paves a square plaza that measures 103 feet on each side. Find the exact area of the plaza in square feet without a calculator or long multiplication. Word problem
- Linnea starts with two identical square carpets that each measure 2x feet on a side, where x is greater than 2. She trims 3 feet off every side of Carpet 1 so it becomes a smaller square of side 2x − 3 feet. On Carpet 2, she adds 3 feet to the length and trims 3 feet from the width so it measures 2x + 3 feet by 2x − 3 feet. Write a polynomial in standard form for the new area of each carpet in square feet. When x = 5, which carpet has the greater area, and by how many square feet? Word problem
- Look at three trinomials: Trinomial A is 9x² + 12x + 4, Trinomial B is 9x² + 6x + 4, and Trinomial C is 4x² − 20x − 25. State which one is a perfect-square trinomial, write it as the square of a binomial, and explain why each of the other two fails the test.
Tier 3 Advanced Several steps - find the hidden structure
- Rami builds a wooden frame of uniform width x inches all the way around a square painting that measures 8 inches on each side. Write a binomial for the outer side length of the framed painting. Then write polynomials in standard form for the total area and for the area of the wooden frame alone, in square inches. Word problem
- Shiori trims 4 centimeters off every side of a square metal plate to make a smaller square plate. The area of the plate decreases by 120 square centimeters. What was the original side length of the plate in centimeters? Word problem
- Efe expands (3x² + 4y)² and writes 6x⁴ + 12x²y + 16y². He made two mistakes on the coefficients. Find both mistakes and write the correct expanded polynomial.
- Mirabel compares the areas of two plots of farmland without a calculator. Plot A is a rectangle measuring 97 meters by 103 meters. Plot B is a square measuring 99 meters on each side. Find the exact area of each plot in square meters. Which plot has the greater area, and by how many square meters? Word problem
- Find both values of the number k that make 16x² + kx + 25 a perfect-square trinomial, and write the trinomial as the square of a binomial for each value of k.
Tier 4 Challenge Stretch problems, some above grade level
- Levon designs a square courtyard of side length 3x + 5 meters with a square fountain of side length 3x − 5 meters centered inside it, where x is a whole number greater than 2. Write the area of the walkway around the fountain as a simplified expression in square meters. Then explain why the walkway's area in square meters is always a multiple of 60. Word problem
- Sanaa expands (x − 2)²(x + 2)². First, she rewrites the expression as ((x − 2)(x + 2))² = (x² − 4)². Second, she expands (x² − 4)² as x⁴ − 16. Explain why her first step is valid, find her mistake in the second step, and write the correct degree-4 polynomial in standard form.
- Let a and b be any real numbers, positive, negative, or zero. Find the simplified difference (a + b)² − (a² + b²). Use that difference to state the exact conditions on a and b for which (a + b)² is greater than a² + b², equal to a² + b², or less than a² + b².
- Iker has two square tiles of side lengths a centimeters and b centimeters. The sum of their side lengths is a + b = 11 centimeters, and the sum of their areas is a² + b² = 73 square centimeters. Without finding a and b, find the exact area ab, in square centimeters, of a rectangle with length a centimeters and width b centimeters. Word problem
- A quadratic polynomial function written in the form y = (x − h)² + k reaches its minimum output value y = k when x = h, because the squared term (x − h)² is never negative and equals 0 at x = h. Parvati flies a drone. Its height y, in meters, x seconds after a timer starts is y = x² − 12x + 41. First, find the number that completes x² − 12x + __ as the perfect-square trinomial (x − 6)², and rewrite x² − 12x + 41 in the form (x − 6)² + k. Then state the time x in seconds when the drone is lowest, and its lowest height y in meters. Word problem Above grade level
Answers
Tier 1 Foundations
- x² + 12x + 36, y² − 8y + 16, and m² − 81. Use (a + b)² = a² + 2ab + b² for the first expression: x² + 2(x)(6) + 6² = x² + 12x + 36. Use (a − b)² = a² − 2ab + b² for the second expression: y² − 2(y)(4) + 4² = y² − 8y + 16. Use (a + b)(a − b) = a² − b² for the third expression: m² − 9² = m² − 81.
- Splitting a square of side x + 5 into sections gives an x-by-x square, a 5-by-5 square, and two x-by-5 rectangles of area 5x each that Callum left out; the correct area is x² + 10x + 25 square inches. A square of side x + 5 has area (x + 5)(x + 5). Using the square-of-a-sum pattern, (x + 5)² = x² + 2(5)(x) + 5². Simplifying gives x² + 10x + 25 square inches, so Callum missed the 10x middle term from the two 5-by-x rectangles.
- Zuleika's combined area is a² + b² square meters and her neighbor's area is a² + 2ab + b² square meters; the neighbor's garden is larger by 2ab square meters. Add the areas of Zuleika's two squares: a² + b² square meters. Expand the neighbor's square of side a + b: (a + b)² = a² + 2ab + b² square meters. Subtract the two expressions: (a² + 2ab + b²) − (a² + b²) = 2ab square meters, which is positive whenever a and b are positive.
- p = 7 and q = 49; r = 6 and s = 12. In (x + p)² = x² + 2px + p², the middle coefficient is 2p = 14, so p = 7 and the constant term is q = 7² = 49. In (y − r)² = y² − 2ry + r², the constant term is r² = 36, so the positive number r is 6. Multiply 2r to find s: s = 2(6) = 12.
- (50 − 1)(50 + 1) = 50² − 1² = 2,499 apples. Write 49 as 50 − 1 and 51 as 50 + 1. Apply the difference-of-squares pattern: (50 − 1)(50 + 1) = 50² − 1². Subtract: 2,500 − 1 = 2,499 apples.
Tier 2 Intermediate
- 9x² − 12xy + 4y² and 16a² − 25b². Apply (A − B)² = A² − 2AB + B² with A = 3x and B = 2y: (3x)² − 2(3x)(2y) + (2y)² = 9x² − 12xy + 4y². Apply (A + B)(A − B) = A² − B² with A = 4a and B = 5b: (4a)² − (5b)² = 16a² − 25b².
- In the first problem, (−3)² is +9, not −9, so (2x − 3)² = 4x² − 12x + 9; in the second problem, a perfect-square trinomial needs the middle term to be twice the product of x and 10, so the middle term would need to be 20x, not 10x. In (2x − 3)², the last term is (−3)² = +9, giving 4x² − 12x + 9. For x² + bx + 100 to equal (x + 10)², the middle term must be 2(x)(10) = 20x. Because 10x does not equal 20x, x² + 10x + 100 is not a perfect-square trinomial.
- 10,609 square feet. Write 103² as (100 + 3)². Apply (a + b)² = a² + 2ab + b²: 100² + 2(100)(3) + 3². Add the three parts: 10,000 + 600 + 9 = 10,609 square feet.
- Carpet 1 has area 4x² − 12x + 9 square feet and Carpet 2 has area 4x² − 9 square feet; when x = 5, Carpet 2 has the greater area by 42 square feet. Expand Carpet 1's area: (2x − 3)² = 4x² − 12x + 9 square feet. Expand Carpet 2's area: (2x + 3)(2x − 3) = 4x² − 9 square feet. At x = 5, Carpet 1 has area (10 − 3)² = 49 square feet and Carpet 2 has area 4(25) − 9 = 91 square feet, so Carpet 2 is greater by 91 − 49 = 42 square feet.
- Trinomial A is a perfect square and equals (3x + 2)²; Trinomial B fails because its middle term is 6x instead of 2(3x)(2) = 12x; Trinomial C fails because its constant term −25 is negative, and a squared real term cannot be negative. In Trinomial A, 9x² = (3x)², 4 = 2², and the middle term is 2(3x)(2) = 12x, so it equals (3x + 2)². In Trinomial B, the middle term 6x is only (3x)(2) instead of 2(3x)(2) = 12x. In Trinomial C, the last term is −25 instead of +25, so it cannot be (2x − 5)².
Tier 3 Advanced
- Outer side length is 2x + 8 inches; total combined area is 4x² + 32x + 64 square inches; frame area alone is 4x² + 32x square inches. Adding a border of width x inches on both sides of the 8-inch painting gives an outer side length of 2x + 8 inches. Expand (2x + 8)² using the square-of-a-sum pattern: (2x)² + 2(2x)(8) + 8² = 4x² + 32x + 64 square inches. Subtract the 8-by-8 painting area of 64 square inches to get 4x² + 32x square inches for the frame alone.
- 17 centimeters. Let x be the original side. The area lost is x² − (x − 4)² = 120. Expand (x − 4)² = x² − 8x + 16. Substitute into the area difference equation and distribute the minus sign: x² − (x² − 8x + 16) = 120, which simplifies to 8x − 16 = 120. Add 16 to get 8x = 136, and divide by 8 to find x = 17 centimeters.
- Efe multiplied 3 by 2 instead of squaring 3 to get 9 on the first term, and he forgot to multiply the cross-product 12x²y by 2; the correct polynomial is 9x⁴ + 24x²y + 16y². Square the first term (3x²)² by squaring 3 and multiplying the exponent by 2: 3² · (x²)² = 9x⁴. Multiply twice the product of the two terms: 2(3x²)(4y) = 24x²y. Square the last term: (4y)² = 16y², giving 9x⁴ + 24x²y + 16y².
- Plot A has area 9,991 square meters and Plot B has area 9,801 square meters; Plot A is greater by 190 square meters. For Plot A, use a difference of squares around 100: 97 × 103 = (100 − 3)(100 + 3) = 100² − 3² = 10,000 − 9 = 9,991 square meters. For Plot B, use a square of a difference around 100: 99² = (100 − 1)² = 100² − 2(100)(1) + 1² = 10,000 − 200 + 1 = 9,801 square meters. Subtract: 9,991 − 9,801 = 190 square meters.
- k = 40 gives (4x + 5)², and k = −40 gives (4x − 5)². The first term is (4x)² and the last term is 5². A perfect-square trinomial can be either (4x + 5)² or (4x − 5)². Expanding gives 16x² + 40x + 25 and 16x² − 40x + 25, so k = 40 or k = −40.
Tier 4 Challenge
- The walkway area simplifies to 60x square meters, which is always 60 times the whole number x. Expand the courtyard area: (3x + 5)² = 9x² + 30x + 25 square meters. Expand the fountain area: (3x − 5)² = 9x² − 30x + 25 square meters. Subtract the fountain area from the courtyard area: (9x² + 30x + 25) − (9x² − 30x + 25) = 60x square meters. Since x is a whole number, 60x is a multiple of 60.
- Her first step is valid by the power-of-a-product exponent rule, but in the second step she treated (x² − 4)² like a difference of squares and missed the middle term −8x²; the correct polynomial is x⁴ − 8x² + 16. By exponent laws, a²b² = (ab)², so (x − 2)²(x + 2)² = ((x − 2)(x + 2))² = (x² − 4)². Apply the square-of-a-difference pattern (A − B)² = A² − 2AB + B² with A = x² and B = 4. Expand: (x²)² − 2(x²)(4) + 4² = x⁴ − 8x² + 16.
- The difference is 2ab; (a + b)² is greater than a² + b² when a and b have the same sign, equal to a² + b² when at least one of a or b is 0, and less than a² + b² when a and b have opposite signs. Expand and subtract: (a² + 2ab + b²) − (a² + b²) = 2ab. When a and b are both positive or both negative, their product ab is positive, so 2ab > 0 and (a + b)² > a² + b². When a = 0 or b = 0, 2ab = 0 so they are equal; when one number is positive and the other is negative, 2ab < 0 so (a + b)² < a² + b².
- 24 square centimeters. Write the square-of-a-sum identity: (a + b)² = a² + 2ab + b² = (a² + b²) + 2ab. Substitute a + b = 11 and a² + b² = 73: 11² = 73 + 2ab, so 121 = 73 + 2ab. Subtract 73 to get 2ab = 48, and divide by 2 to find ab = 24 square centimeters.
- The constant 36 completes the square, giving y = (x − 6)² + 5; the drone is lowest at 6 seconds, at a height of 5 meters. Expand (x − 6)² using the square-of-a-difference pattern: (x − 6)² = x² − 12x + 36, so 36 completes the perfect-square trinomial. Split 41 as 36 + 5: y = (x² − 12x + 36) + 5 = (x − 6)² + 5. Since (x − 6)² ≥ 0 for all real x and equals 0 at x = 6, the lowest height is y = 0 + 5 = 5 meters at x = 6 seconds.
Stuck on this topic?
Trouble here often traces back to an earlier skill. Our free check-in finds the exact gap. A tutor then works on it with your child.
Find gaps free More practice