pAIrmatic Lessons
UNIT 1PolynomialsLesson 3 of 31 Adding, Subtracting and Multiplying Polynomials2 Special Products3 Factoring: GCF and Grouping

Factoring: GCF and Grouping

A1.6Algebra 1Algebra20 practice questions

Factoring undoes multiplying. You write a polynomial as a product of simpler parts.

What you'll be able to do

The big idea

The GCF. Take the greatest common factor of the numbers. Then take each shared letter with its lowest exponent. The GCF of 12x³y and 18x²y² is 6x²y.

Divide every term. 10x³ − 15x² = 5x²(2x − 3). Each term inside is a term of the polynomial divided by 5x².

Leave a 1. If a whole term is the GCF, a 1 is left in its place: 4y³ + 4y = 4y(y² + 1).

A negative GCF. If the first term is negative, take out a negative. Every sign inside flips: −3x² + 12x = −3x(x − 4).

A binomial can be the GCF. In 2x(x + 5) + 3(x + 5), both parts share (x + 5). So it equals (x + 5)(2x + 3).

Grouping. For four terms, factor the first pair and the last pair. If both leave the same binomial, take it out: x³ + 2x² + 4x + 8 = x²(x + 2) + 4(x + 2) = (x + 2)(x² + 4).

Check by multiplying. Multiply your factors. You should get back the polynomial you started with.

Words you need

Factor
A part you multiply. To factor means to write something as a product.
Greatest common factor
The GCF. The biggest factor that every term shares. The GCF of 12 and 18 is 6.
Product
The answer when you multiply.
Polynomial
A sum of terms, like 3x³ − 6x² + 5x − 10.
Term
One part of a polynomial, like 5x or −10.
Monomial
A polynomial with 1 term, like 6x²y.
Binomial
A polynomial with 2 terms, like x + 5.
Trinomial
A polynomial with 3 terms, like 2a − 3b + 1.
Grouping
Factoring four terms in two pairs, then taking out the binomial both pairs share.
Completely factored
No factor can be factored any more.
Base
The bottom of a shape, like the bottom of a box. In a power like 3², the base is 3.
Coefficient
The number in front of a letter. In 7x², it is 7.
Constant
A term with no letter, like −10.
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.
Zero
An x-value that makes a function equal 0.

Grouping, step by step

StepResult
Start3x³ − 6x² + 5x − 10
Make two pairs(3x³ − 6x²) + (5x − 10)
Factor each pair3x²(x − 2) + 5(x − 2)
Take out the shared binomial(x − 2)(3x² + 5)
Check by multiplying3x³ + 5x − 6x² − 10

Worked examples

Example 1

Factor 14a³b² − 21a²b³ + 7a²b² completely.

  1. The GCF of 14, 21 and 7 is 7. The lowest powers are a² and b². So the GCF is 7a²b².
  2. Divide each term by 7a²b²: 2a, −3b and 1.
  3. The answer is 7a²b²(2a − 3b + 1). The 1 is there because the last term was the GCF itself.

Answer: 7a²b²(2a − 3b + 1)

Example 2

Factor −8x³ + 20x². Make the first term inside the brackets positive.

  1. The first term is negative, so take out a negative GCF: −4x².
  2. Divide each term by −4x²: −8x³ gives 2x, and 20x² gives −5.
  3. The answer is −4x²(2x − 5). Check: −4x² · 2x = −8x³ and −4x² · (−5) = 20x².

Answer: −4x²(2x − 5)

Example 3

Factor 2x³ − 18 + 3x² − 12x.

  1. As written, the pairs 2x³ − 18 and 3x² − 12x share no binomial.
  2. Put the terms in standard form: 2x³ + 3x² − 12x − 18.
  3. Factor each pair: x²(2x + 3) − 6(2x + 3).
  4. Take out (2x + 3): (2x + 3)(x² − 6).

Answer: (2x + 3)(x² − 6)

Common mistakes

Dropping the 1. 5x² + 5x = 5x(x + 1), not 5x(x).

Stopping before the GCF is the greatest. 12x² + 8x = 2x(6x + 4) is not done, because 6x + 4 still shares 2. Write 4x(3x + 2).

Missing a sign with a negative GCF. −2x² − 6x = −2x(x + 3), not −2x(x − 3).

Losing the minus sign before a pair. In x³ + 3x² − 2x − 6, take −2 out of the last pair: x²(x + 3) − 2(x + 3).

Practice: 20 questions

13 of 20 are word problems. For a word problem, sketch it first and mark what you know.

Tier 1 Foundations Use the rule directly

  1. Find the greatest common factor of the three terms 18x⁴y², 24x³y³, and 30x²y⁴. Then factor the polynomial 15a³b − 25a²b² completely.
  2. Alaric designs a rectangular solar panel whose area is 8x² + 12x square feet. The width of the panel in feet is the greatest common factor of the two area terms. Factor the area polynomial to find expressions for the width and the length of the panel in feet, and find both dimensions when x = 3. Word problem
  3. Belen makes a rectangular hallway runner whose area is 6x² + 6x square feet and whose width is 6x feet. She factors out 6x to find the length and writes 6x² + 6x = 6x(x), concluding that the length is x feet. Explain how checking her answer by multiplying shows her mistake, and write the correct factored form and length in feet. Word problem
  4. Chidi builds two rectangular garden beds, where x is a positive whole number. Bed A has an area of 12x² + 18x square meters, and Bed B has an area of 15x² + 10x square meters. The width of each bed is the greatest common factor of its area terms. Factor both areas to find the width and length of each bed in meters. For every positive whole number x, which bed has the greater width, and which bed has the greater length? Word problem
  5. Factor each expression by factoring out the common binomial: Expression A is 4x(x − 5) + 7(x − 5), and Expression B is 2y(3y + 1) − (3y + 1).

Tier 2 Intermediate One more step to think through

  1. Dilara models the height y, in meters, of a fountain's water arc at horizontal distance x meters from the nozzle with the polynomial y = −4x² + 20x. Factor out the negative greatest common factor so that the leading term inside the parentheses is positive, and find the height of the water arc at x = 3 meters. Word problem
  2. Enitan completes two factoring problems. First, he factors 8x² + 4x as 2x(4x + 2) and marks it complete. Second, he factors −6x² − 15x by taking out −3x and writes −3x(2x − 5). Find his mistake in each problem and write the correct completely factored form for both.
  3. Factor 2x³ + 6x² − 5x − 15 completely by grouping, and show how multiplying your two binomial factors checks your answer.
  4. Fiorella compares two rectangular solar panels whose areas in square feet are written as four-term polynomials, where x is a positive number. Panel A has area x² + 4x + 3x + 12 square feet, and Panel B has area x² + 6x + 2x + 12 square feet. Without combining the middle terms, factor each four-term polynomial by grouping to find the length and width of each panel in feet. Which panel has the greater perimeter for every positive value of x, and by how many feet? Word problem
  5. Goran builds a rectangular wooden planter whose volume is 12x³ + 18x² + 6x cubic inches. The height of the planter in inches is the greatest common factor of the three volume terms. Factor out the GCF to find the height in inches and the trinomial representing the base area in square inches. Word problem

Tier 3 Advanced Several steps - find the hidden structure

  1. In the polynomial 6x³ − 10 + 4x² − 15x, the first two terms and the last two terms do not share a common binomial factor as written. Reorder the terms and factor the polynomial completely by grouping.
  2. Hina divides a rectangular courtyard into four tiled zones whose areas add up to 3xy + 8 + 12x + 2y square meters. Reorder the terms and factor by grouping to find binomial expressions for the length and width of the courtyard in meters, and find the perimeter of the courtyard when x = 3 and y = 5. Word problem
  3. Imre factors 4x³ + 8x² − 3x − 6 by grouping. He writes 4x²(x + 2) − 3(x + 2), and then writes his final answer as 4x²(x + 2). Show by multiplying why 4x²(x + 2) does not equal the original polynomial, explain what Imre did wrong, and write the correct factored form.
  4. Jendayi builds two rectangular tabletops, where x is greater than 2. Tabletop A has an area of 6x² + 9x − 4x − 6 square inches, and Tabletop B has an area of 6x² − 9x + 4x − 6 square inches. Without combining the middle terms, factor both four-term polynomials by grouping to find the two side lengths of each tabletop in inches. Which tabletop has the greater perimeter, and by how many inches? Word problem
  5. Kirill designs a rectangular storage container whose volume is 2x⁴ + 6x³ + 10x² + 30x cubic feet. Factor the volume completely. Write the three factors as the dimensions of the container in feet. Word problem

Tier 4 Challenge Stretch problems, some above grade level

  1. Laleh lays out a rectangular warehouse floor divided into four sections whose combined area is 6x³ + 15x² + 4x + k square meters. Find the number k that allows this four-term polynomial to be factored by grouping, write the factored binomial expressions for the width and length of the floor in meters, and find the area of the floor when x = 2. Word problem
  2. Makoa tries to factor 5x(2x − 3) + 4(3 − 2x). He says that because 2x − 3 and 3 − 2x are different binomials, the expression cannot be factored any further. Explain how 3 − 2x is related to 2x − 3, correct Makoa's claim, and write the completely factored expression.
  3. Nori designs two rectangular courtyards whose areas in square meters are written as four-term polynomials, where x > y > 0. Courtyard A has area 2x² + 2xy + 3xy + 3y² square meters, and Courtyard B has area 2x² + 6xy + xy + 3y² square meters. Without combining the xy terms, factor each four-term expression by grouping to find the length and width of each courtyard in meters. Does either courtyard have a greater perimeter? Explain. Word problem
  4. Orestes pours concrete for a rectangular retaining wall whose volume is 6x⁴ − 30x + 15x³ − 12x² cubic feet, where x is greater than 2. Factor the volume completely into three factors for the thickness, height, and length of the wall in feet. Find the three dimensions when x = 3. Word problem
  5. A zero of a polynomial function is an x-value that makes the output y equal 0, and a product of factors equals 0 when at least one factor equals 0. Pema fills a tank. The water level y, in centimeters above a mark on the tank, x minutes after she starts is y = 2x³ − 6x² + 5x − 15. A negative y means the water is below the mark. Factor the polynomial by grouping into two factors, explain why one of the two factors can never equal 0 for any real number x, and find the only real zero of the function. Word problem Above grade level

Answers

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