UNIT 1PolynomialsLesson 6 of 61 Adding, Subtracting and Multiplying Polynomials2 Special Products3 Factoring: GCF and Grouping4 Factoring x² + bx + c5 Factoring ax² + bx + c6 Factoring Special Cases
Factoring Special Cases
Some polynomials match a pattern. Spot the pattern, and you can factor them in one step.
What you'll be able to do
- Factor a difference of squares.
- Factor a perfect square trinomial.
- Take out a GCF first, and factor more than once.
- Tell which method fits, and when a sum of squares can't be factored.
The big idea
Difference of squares. a² − b² = (a + b)(a − b). So 9x² − 4 = (3x + 2)(3x − 2).
Perfect square trinomials. a² + 2ab + b² = (a + b)², and a² − 2ab + b² = (a − b)². Check that the middle term is 2ab.
GCF first. 5x² − 20 = 5(x² − 4) = 5(x + 2)(x − 2).
Factor again. x⁴ − 16 = (x² + 4)(x² − 4). The part x² − 4 is a difference of squares too.
A sum of squares. x² + 4 can't be factored with real numbers. Two numbers that multiply to +4 have the same sign, so their sum is not 0.
Mental math. 51² − 49² = (51 + 49)(51 − 49) = 100 × 2 = 200.
Words you need
- Difference of squares
- a² − b², one square minus another. It equals (a + b)(a − b).
- Perfect square trinomial
- A trinomial that equals a binomial squared, like x² − 14x + 49 = (x − 7)².
- Sum of squares
- a² + b², like x² + 9. It can't be factored with real numbers.
- Perfect square
- A number or term that is something times itself, like 49 or 9x².
- Square root
- A number that gives the first number when you multiply it by itself. √49 = 7.
- Factor
- A part you multiply. To factor means to write something as a product.
- Factor completely
- Keep factoring until no factor can be factored any more.
- Product
- The answer when you multiply.
- Trinomial
- A polynomial with 3 terms.
- Binomial
- A polynomial with 2 terms, like 3x + 2.
- Polynomial
- A sum of terms, like x⁴ − 16.
- Term
- One part of a polynomial, like 9x² or −4.
- Coefficient
- The number in front of a letter. In 9x², it is 9.
- Constant
- A term with no letter, like −4.
- Greatest common factor
- The GCF. The biggest factor that every term shares.
- Integer
- A whole number or its negative, like −3, 0 or 7.
- Real number
- Any number on the number line.
- Exponent
- The small raised number. In x⁴, it is 4.
- Power
- A base with an exponent, like x⁴.
- Base
- The bottom of a shape, like the bottom of a box. In a power like 3², the base is 3.
- 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.
- Zero
- An input value that makes a function equal 0.
- Perfect cube
- A number or term that is something times itself 3 times, like 27 or 8x³.
Which pattern?
| Form | Factors | How to spot it |
|---|---|---|
| a² − b² | (a + b)(a − b) | Two squares with a minus between them |
| a² + 2ab + b² | (a + b)² | Squares at both ends, middle term +2ab |
| a² − 2ab + b² | (a − b)² | Squares at both ends, middle term −2ab |
| a² + b² | Can't be factored with real numbers | Two squares with a plus between them |
Worked examples
Example 1
Factor 25x² − 9.
- 25x² = (5x)² and 9 = 3². There is a minus between them.
- So it is a difference of squares with a = 5x and b = 3.
- 25x² − 9 = (5x + 3)(5x − 3).
Answer: (5x + 3)(5x − 3)
Example 2
Factor x² − 14x + 49.
- x² = (x)² and 49 = 7².
- The middle term should be 2 · x · 7 = 14x. It is −14x, so the sign is minus.
- x² − 14x + 49 = (x − 7)².
Answer: (x − 7)²
Example 3
Factor 2x² − 72 completely.
- Every term shares 2. Take it out: 2(x² − 36).
- x² − 36 is a difference of squares: (x + 6)(x − 6).
- So 2x² − 72 = 2(x + 6)(x − 6).
Answer: 2(x + 6)(x − 6)
Example 4
Factor x⁴ − 1 completely.
- x⁴ = (x²)² and 1 = 1². So x⁴ − 1 = (x² + 1)(x² − 1).
- x² − 1 is a difference of squares again: (x + 1)(x − 1).
- x² + 1 is a sum of squares, so it stays.
- So x⁴ − 1 = (x² + 1)(x + 1)(x − 1).
Answer: (x² + 1)(x + 1)(x − 1)
Common mistakes
Squaring instead of splitting. x² − 25 is (x + 5)(x − 5), not (x − 5)². (x − 5)² = x² − 10x + 25.
Factoring a sum of squares. x² + 25 is not (x + 5)(x − 5). That product is x² − 25.
Stopping too soon. (x² + 9)(x² − 9) is not done. Factor x² − 9 into (x + 3)(x − 3).
Missing the GCF. 2x² − 18 is not a difference of squares as it stands. Take out 2 first: 2(x + 3)(x − 3).
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
- Factor 16x² − 49 and 81 − 4y². Check your answers by multiplying the binomials.
- A square tabletop has an area of 9x² + 30x + 25 square inches. Explain how you can tell from the terms that this polynomial is a perfect square trinomial, and write it in factored form to find the side length of the tabletop in inches. What is the side length in inches when x = 4? Word problem
- A square garden of side x meters has a square tool shed of side 4 meters in one corner, where x is greater than 4. The remaining garden area is x² − 16 square meters. Cassian factors x² − 16 as (x − 4)². Explain how multiplying (x − 4)² shows his mistake, and write the correct factored form. Word problem
- Look at the expressions 4x² − 12x + 9, 4x² − 6x + 9, and 4x² + 12x − 9. Decide which of these expressions are perfect square trinomials, write each perfect square trinomial in factored form, and explain why any remaining expression is not a perfect square trinomial.
- A public plaza consists of a square outer courtyard of side 65 feet with a square fountain of side 35 feet in the center. The paved walking area around the fountain is 65² − 35² square feet. Without a calculator, factor the expression to find the exact paved area in square feet. Word problem
Tier 2 Intermediate One more step to think through
- A rectangular wooden beam has a volume of 3x³ − 75x cubic inches, where x is greater than 5. Factor the volume polynomial completely to find expressions for the thickness, width, and length of the beam in inches. What are those dimensions in inches when x = 7? Word problem
- Explain why x² − 16 can be factored with real numbers while x² + 16 cannot. Support your explanation by multiplying out (x + 4)(x − 4), (x + 4)², and (x − 4)².
- Yvette writes that x² + 9 factors as (x + 3)(x − 3), and that 9x² − 24x + 16 factors as (3x + 4)(3x − 4). Explain her mistake in each problem, and give the correct classification or factored form for each polynomial.
- A park has a square courtyard and a rectangular lawn, where x is greater than 2. The courtyard has an area of 9x² + 24x + 16 square meters. The lawn has an area of 9x² − 16 square meters. Factor each polynomial to find the side lengths of the courtyard and the lawn in meters. Compare the perimeter of the courtyard and the perimeter of the lawn. Which space has the greater perimeter, and by how many meters? Word problem
- A square metal plate has a side length of 5x centimeters. A machinist cuts a square hole of side length 2y centimeters out of the center of the plate, where 5x is greater than 2y. Write a polynomial for the remaining area of the plate in square centimeters, and write it in factored form. Find the remaining area in square centimeters when x = 6 and y = 5. Word problem
Tier 3 Advanced Several steps - find the hidden structure
- Factor each polynomial completely and state which factoring method or special pattern applies after any greatest common factor is removed: Polynomial A is 2x² − 12x + 18, Polynomial B is 6x² + 7x − 20, and Polynomial C is 5x³ − 45x.
- Osman factors x⁴ − 16 as (x² + 4)(x² − 4) and marks his work complete. Explain why his factorization is not complete, and write the completely factored form of x⁴ − 16.
- A stone dropped from a bridge over a gorge has a height, in feet above the river after t seconds, modeled by −16t² + 144. Write this polynomial in completely factored form with a negative leading factor. What is the height of the stone in feet at t = 2 seconds? Word problem
- Two public plazas have decorative stone sections. Section A has an area of 16x⁴ − 81 square feet. Section B has an area of 16x⁴ − 72x² + 81 square feet. Factor each polynomial completely over the integers. When any squared binomial is written out as repeated binomial factors, compare the total number of binomial factors in Section A's factored form and Section B's factored form. Word problem
- Sabine tiles a square hotel lobby that measures 84 feet on each side, except for a square sunken seating area in the center that measures 16 feet on each side. The tiled floor area is 84² − 16² square feet, and the stone tile costs $3 per square foot. Without a calculator, factor the area expression to find the exact tiled area in square feet and the total cost of the tile in dollars. Word problem
Tier 4 Challenge Stretch problems, some above grade level
- A square park has a side length of 3x + 5 meters. Inside the park is a square dog run of side length x − 1 meters, where x is greater than 2. The grass area outside the dog run is (3x + 5)² − (x − 1)² square meters. Factor this difference of squares and simplify completely so that any common numerical factors are pulled to the front. What is the grass area in square meters when x = 5? Word problem
- Linnea factors 81x⁴ − 1. In Step 1, she writes (9x² − 1)(9x² − 1). In Step 2, she factors each binomial to get (3x + 1)(3x − 1)(3x + 1)(3x − 1). Find her mistake in Step 1, explain why one of the true factors in Step 1 cannot be factored further over the real numbers, and write the correct completely factored form.
- Factor the three polynomials x⁴ − 16, x⁴ − 8x² + 16, and x⁴ + 8x² + 16 completely over the integers. Compare their completely factored forms by stating which polynomial breaks down entirely into first-degree binomial factors and which polynomial contains no first-degree binomial factors at all.
- A wooden shipping crate has a volume of x³ + 4x² − 9x − 36 cubic feet, where x is greater than 3. Factor this polynomial completely into linear binomials to find expressions for the dimensions of the crate in feet. What are the dimensions of the crate in feet when x = 5? Word problem
- A zero of a polynomial function is an input value that makes the output equal 0, and a product of factors equals 0 when any factor equals 0. Consider the polynomial function y = x⁴ − 13x² + 36. Factor it completely into linear binomials, and find every real zero of the function. Above grade level
Answers
Tier 1 Foundations
- (4x + 7)(4x − 7) and (9 + 2y)(9 − 2y). Rewrite 16x² − 49 as (4x)² − 7², which factors into (4x + 7)(4x − 7). Rewrite 81 − 4y² as 9² − (2y)², which factors into (9 + 2y)(9 − 2y). Multiply to check: (4x + 7)(4x − 7) = 16x² − 49, and (9 + 2y)(9 − 2y) = 81 − 4y².
- The first term is (3x)², the last term is 5², and the middle term 30x is twice the product of 3x and 5; the factored form is (3x + 5)², so the side length is 3x + 5 inches; when x = 4, the side length is 17 inches. Check the outer terms: 9x² = (3x)² and 25 = 5². Check the middle term: 2(3x)(5) = 30x, which matches +30x, so 9x² + 30x + 25 = (3x + 5)². When x = 4, the side length is 3(4) + 5 = 17 inches.
- Multiplying (x − 4)² gives x² − 8x + 16, which has a −8x middle term and +16 instead of −16; the correct factored form is (x + 4)(x − 4). Expand Cassian's expression: (x − 4)(x − 4) = x² − 4x − 4x + 16 = x² − 8x + 16. For the middle terms to cancel and the constant term to be −16, the signs in the two binomials must be opposite. Thus x² − 16 factors as (x + 4)(x − 4).
- 4x² − 12x + 9 is the only perfect square trinomial and factors as (2x − 3)²; 4x² − 6x + 9 fails because its middle term is −6x instead of −2(2x)(3) = −12x, and 4x² + 12x − 9 fails because its constant term −9 is negative. In 4x² − 12x + 9, 4x² = (2x)², 9 = 3², and −12x = −2(2x)(3), so it equals (2x − 3)². In 4x² − 6x + 9, the middle term is missing the factor of 2 needed for −2(2x)(3) = −12x. In 4x² + 12x − 9, the last term is −9, and the square of a real number cannot be negative.
- (65 + 35)(65 − 35) = 100 × 30 = 3,000 square feet. Apply the difference-of-squares pattern to 65² − 35²: (65 + 35)(65 − 35). Add and subtract inside the parentheses: 65 + 35 = 100 and 65 − 35 = 30. Multiply: 100 × 30 = 3,000 square feet.
Tier 2 Intermediate
- 3x(x + 5)(x − 5); when x = 7, the dimensions are 21 inches, 12 inches, and 2 inches. Factor out the greatest common factor 3x from both terms: 3x(x² − 25). Factor the difference of squares x² − 25 into (x + 5)(x − 5), giving 3x(x + 5)(x − 5). When x = 7, the dimensions are 3(7) = 21 inches, 7 + 5 = 12 inches, and 7 − 5 = 2 inches.
- (x + 4)(x − 4) = x² − 16 because +4x and −4x cancel while (+4)(−4) = −16; however, for two numbers to multiply to +16 they must have the same sign, which produces a non-zero middle term of +8x in (x + 4)² = x² + 8x + 16 or −8x in (x − 4)² = x² − 8x + 16, so x² + 16 is prime over the real numbers. Multiply (x + 4)(x − 4) = x² − 4x + 4x − 16 = x² − 16. Multiply (x + 4)² = x² + 8x + 16 and (x − 4)² = x² − 8x + 16. Any real numbers p and q with pq = 16 have the same sign, so p + q cannot equal 0, making x² + 16 impossible to factor with real numbers.
- Multiplying (x + 3)(x − 3) gives x² − 9, not x² + 9, and x² + 9 is a sum of squares that cannot be factored with real numbers; multiplying (3x + 4)(3x − 4) gives 9x² − 16, missing −24x, and 9x² − 24x + 16 is a perfect square trinomial that factors as (3x − 4)². Check the first problem: (x + 3)(x − 3) = x² − 9. Because x² + 9 is a sum of squares, it is prime over the real numbers. Check the second problem: (3x + 4)(3x − 4) = 9x² − 16, which has no middle term. Since 9x² = (3x)², 16 = 4², and −24x = −2(3x)(4), 9x² − 24x + 16 factors as (3x − 4)².
- The courtyard has side lengths 3x + 4 meters and 3x + 4 meters; the lawn has side lengths 3x + 4 meters and 3x − 4 meters; the courtyard has the greater perimeter by 16 meters. Factor the courtyard's area: 9x² + 24x + 16 = (3x + 4)², so its perimeter is 4(3x + 4) = 12x + 16 meters. Factor the lawn's area: 9x² − 16 = (3x + 4)(3x − 4), so its perimeter is 2(3x + 4 + 3x − 4) = 12x meters. Subtract the perimeters: (12x + 16) − 12x = 16 meters.
- 25x² − 4y² = (5x + 2y)(5x − 2y) square centimeters; 800 square centimeters when x = 6 and y = 5. Subtract the hole's area from the plate's area: (5x)² − (2y)² = 25x² − 4y² square centimeters. Factor the difference of squares: (5x + 2y)(5x − 2y). When x = 6 and y = 5, 5x = 30 and 2y = 10, so the area is (30 + 10)(30 − 10) = 40 × 20 = 800 square centimeters.
Tier 3 Advanced
- Polynomial A is 2(x − 3)² by GCF and perfect square trinomial; Polynomial B is (2x + 5)(3x − 4) by grouping or ac method; Polynomial C is 5x(x + 3)(x − 3) by GCF and difference of squares. In Polynomial A, factor out 2 to get 2(x² − 6x + 9), which is a perfect square trinomial: 2(x − 3)². In Polynomial B, no GCF > 1 exists and 6x² + 7x − 20 is not a special product; since 15 × (−8) = −120 and 15 + (−8) = 7, group 6x² + 15x − 8x − 20 = 3x(2x + 5) − 4(2x + 5) = (2x + 5)(3x − 4). In Polynomial C, factor out 5x to get 5x(x² − 9), and factor the difference of squares as 5x(x + 3)(x − 3).
- The factor x² − 4 is itself a difference of squares and can be factored further into (x + 2)(x − 2), whereas x² + 4 is a sum of squares and cannot be factored further with real numbers; the completely factored form is (x² + 4)(x + 2)(x − 2). Rewrite x⁴ − 16 as (x²)² − 4² = (x² + 4)(x² − 4). Check each factor: x² + 4 is a sum of squares, so it cannot be factored further over the real numbers. Factor the difference of squares x² − 4 into (x + 2)(x − 2) to get (x² + 4)(x + 2)(x − 2).
- −16(t + 3)(t − 3); 80 feet at t = 2 seconds. Factor out the negative greatest common factor −16 from both terms: −16(t² − 9). Factor the difference of squares t² − 9 into (t + 3)(t − 3), giving −16(t + 3)(t − 3). At t = 2, the height is −16(2 + 3)(2 − 3) = −16(5)(−1) = 80 feet.
- Section A factors as (4x² + 9)(2x + 3)(2x − 3), which is a product of 3 binomial factors; Section B factors as (2x + 3)²(2x − 3)², which is a product of 4 binomial factors when written out, so Section B has 1 more binomial factor than Section A. Factor Section A: 16x⁴ − 81 = (4x² + 9)(4x² − 9) = (4x² + 9)(2x + 3)(2x − 3), giving 3 binomial factors because 4x² + 9 is a sum of squares. Factor Section B as a perfect square trinomial first: 16x⁴ − 72x² + 81 = (4x² − 9)². Factor 4x² − 9 inside the square: ((2x + 3)(2x − 3))² = (2x + 3)²(2x − 3)², which is 4 linear binomial factors when written out.
- Tiled area is 6,800 square feet; total cost is $20,400. Factor 84² − 16² as a difference of squares: (84 + 16)(84 − 16). Simplify each factor: 100 × 68 = 6,800 square feet. Multiply by $3 per square foot: 6,800 × $3 = $20,400.
Tier 4 Challenge
- 8(x + 1)(x + 3) square meters; 384 square meters when x = 5. Apply the difference-of-squares pattern A² − B² = (A + B)(A − B) with A = 3x + 5 and B = x − 1: ((3x + 5) + (x − 1))((3x + 5) − (x − 1)). Simplify inside each set of parentheses: (4x + 4)(2x + 6). Factor 4 out of 4x + 4 and 2 out of 2x + 6 to get 8(x + 1)(x + 3) square meters. At x = 5, the area is 8(5 + 1)(5 + 3) = 8(6)(8) = 384 square meters.
- In Step 1, 81x⁴ − 1 is a difference of squares (9x²)² − 1², which factors as (9x² + 1)(9x² − 1), not (9x² − 1)(9x² − 1); because 9x² + 1 is a sum of squares, it cannot be factored further with real numbers, so the completely factored form is (9x² + 1)(3x + 1)(3x − 1). Rewrite 81x⁴ − 1 as (9x²)² − 1² and factor as a difference of squares: (9x² + 1)(9x² − 1). The factor 9x² + 1 is a sum of squares and cannot be factored further over the real numbers. Factor the difference of squares 9x² − 1 into (3x + 1)(3x − 1) to get (9x² + 1)(3x + 1)(3x − 1).
- x⁴ − 16 = (x² + 4)(x + 2)(x − 2), x⁴ − 8x² + 16 = (x + 2)²(x − 2)², and x⁴ + 8x² + 16 = (x² + 4)²; x⁴ − 8x² + 16 breaks down entirely into first-degree binomial factors, while x⁴ + 8x² + 16 contains no first-degree binomial factors. Factor x⁴ − 16 as a difference of squares: (x² + 4)(x² − 4) = (x² + 4)(x + 2)(x − 2). Factor x⁴ − 8x² + 16 as a perfect square trinomial and then a difference of squares: (x² − 4)² = ((x + 2)(x − 2))² = (x + 2)²(x − 2)², which consists entirely of first-degree factors. Factor x⁴ + 8x² + 16 as a perfect square trinomial: (x² + 4)². Since x² + 4 is a sum of squares, it has no first-degree factors over the real numbers.
- (x + 4)(x + 3)(x − 3); when x = 5, the dimensions are 9 feet, 8 feet, and 2 feet. Group the first two terms and the last two terms: x²(x + 4) − 9(x + 4) = (x + 4)(x² − 9). Factor the difference of squares x² − 9 into (x + 3)(x − 3), giving (x + 4)(x + 3)(x − 3). At x = 5, the dimensions are 5 + 4 = 9 feet, 5 + 3 = 8 feet, and 5 − 3 = 2 feet.
- Factored form is y = (x + 2)(x − 2)(x + 3)(x − 3); the real zeroes are x = −3, x = −2, x = 2, and x = 3. Treat x² as the variable unit: since −4 × (−9) = 36 and −4 + (−9) = −13, x⁴ − 13x² + 36 factors as (x² − 4)(x² − 9). Factor each difference of squares completely: (x + 2)(x − 2)(x + 3)(x − 3). Set each factor equal to 0 to find the real zeroes: x = −2, x = 2, x = −3, and x = 3.
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