Introduction to Factoring Polynomials

Introduction to Factoring Polynomials

Multiple Choice

Find the greatest common factor of 24, 64, 108.

• 2

• 4

• 8

• 24

Multiple Choice

What is the greatest common factor of the monomials a³b³ and a²b⁵?

• a²b³

• a²b⁵

• a³b³

• a³b⁵

Multiple Choice

What is the greatest common factor of the polynomial 30w³ + 48w + 12w²?

• 6w

• 6w³    

• 12w

• 12w³

Multiple Choice

What is the factored form of 56a³ – 8a?

• 8a (7a³ – a)

• 8a² (35a² – a)

• 8a (7a² – 1)

• 8a² (56a³ – 8a)

Multiple Choice

Complete the factor of 7x²y⁵ + 56x²y⁴ = 7x²y⁴ ( ______ ).

• y + 8

• 7y + 8x

• y + 8x²

• x²y + 8

Multiple Choice

Multiply (25 – 7) (25 + 7).

• 625

• 576    

• 484

• 49

Multiple Choice

Which of the following polynomials can be factored using the difference of two squares formula?

• x² + 16

• 25 – y²

• x³ + 1

• x³ – 27

Multiple Choice

When factored, the expression 16x² – 25y² is equivalent to __.

• (4x – 5y) (4x + 5y)

• (8x – 5y) (8x + 5y)

• (4x – 5y) (4x - 5y)

• (8x – 5y) (8x - 5y)

Multiple Choice

Factor 16x² – 100.

• (2x – 5) (2x + 5)

• 4(2x – 10) (2x + 10)

• 4(2x – 5) (2x + 5)

• 4(4x – 5) (4x + 5)

Multiple Choice

Factor x⁴ – y⁴ completely.

• (x² – y²) (x² – y²)

• (x² – y²) (x² + y²)

• (x – y) (x + y)

  (x² + y²)

• (x – y) (x + y)

  (x² - y²)

​What's In

​When you were in elementary, you have already learned about factoring.

POLYNOMIALS WITH COMMON MONOMIAL FACTOR

​Find the Greatest Common Factor (GCF) of the following numbers

Multiple Choice

18 and 36

• 2

• 6

• 9

• 18

Multiple Choice

30 and 42

• 2

• 6

• 15

• 18

Multiple Choice

42 and 49

• 2

• 3

• 6

• 7

Multiple Choice

16, 40, and 56

• 2

• 4

• 8

• 16

Multiple Choice

42, 49, and 70

• 7

• 14

• 21

• 28

Multiple Choice

42 and 56

• 2

• 3

• 7

• 8

Multiple Choice

37 and 51

• 3

• 9

• 17

• none of these

Multiple Choice

49 and 77

• 3

• 7

• 9

• none of these

Multiple Choice

64, 48 and 16

• 4

• 8

• 16

• none of these

Multiple Choice

48, 64 and 32

• 4

• 8

• 16

• none of these

Identify the factors and the product in each problem.

​Example:

​

​4(5) = 20                           factors: 4,5               product: 20

​

5(2 x²) = 10x²                   factors: 5, 2 x²          product: 10x²

Multiple Choice

2(7x²y) = 14x²y

• factors: 2(7x²y)

  product: 14x²y

• factors: (7x²y)

  product: 14x²y

• factors: 2(7x²y)

  product: 4x²y

Multiple Choice

3a(7a²b³) = 12a³b³

• factors: 3a(6a²b³)

  product: 112a³b³

• factors: 3a(7a²b³)

  product: 12a³b³

• factors: a(7a²b³)

  product: 12a³b³

Multiple Choice

-5b³c²(11c²) = -55b³c⁴

• factors: -5b³c²(11c²)

  product: 55b³c⁴

• factors: 5b³c²(11c²)

  product: -55b³c⁴

• factors: -5b³c²(11c²)

  product: -55b³c⁴

Multiple Choice

13m³n(3mn) = 39m⁴n²

• factors: 3m³n(3mn)

  product: 39m⁴n²

• factors: 13m³n(3mn)

  product: 39m⁴n²

• factors: 13m³n(mn)

  product: 39m⁴n²

Multiple Choice

-17(-3klm²) = 51klm²

• factors: -17(-3klm²)

  product: 15klm²

• factors: -17(-3klm²)

  product: 51klm²

• factors: -17(-3klm²)

  product: 151klm²

​Angkan-Angkan Festival or Rehiyon-Rehiyon

Marikina City is known for its different festives like the “Angkan-Angkan.” Are you familiar with this annual event? This festivity  instills -the values of length of connection and solidarity and is celebrated for seven days with the theme of “Ka-angkan Ko, Mabuting Tao.”, More so, festivity showcased more advances with regard to the qualities and great characteristics of Marikeños.

If you have watched any parade like Angkan-Angkan, Rehiyon-Rehiyon, and others, then answer the following:

1.    Can you describe how they were arranged or organized?

2.    What are the things that are common to the parade that you have watched?

3.    Can you identify things that you observed common in the parade?

4.    Why do you think it is useful to find what is common to two or more things?

5.    Why is it important to know the commonality of the different things around us?

​***Now, if you are one of the organizers of the said festival or activity, what aspect/s in the parade will you consider for a much better output or result of the program?

​What is It

A common method of factoring numbers is to completely factor the number into positive prime factors. A prime number is a number whose only positive factors are 1 and itself.

​In polynomials, the first method for factoring will be factoring out the greatest common factor. This is generally the first thing that we should try as it will often simplify the problem.

​ But prior to that, you have to recall these: A monomial is a type of polynomial expression that is the product of constants and nonnegative integer powers of variables, like 2, −4x² , abc, and −2e²f³g⁵. While the other types of polynomials are binomial, trinomial, and multinomial.

​

And to factor a polynomial with common monomial expressions, first, we have to factor the numerical coefficient into positive prime factors completely. Simply write the complete factorization of each monomial and find the common factors.

​Example 1:  

The GCF of 12, 18, and 36.

The GCF is 6.

​Example 2:   The GCF of 10x³ and 4x.

        10x³ = 2⋅5⋅x⋅x⋅x           

          4x = 2⋅2⋅x

​

Therefore, the GCF is 2x

Example 3:   Factor 6ab + 18bc

                      6ab = 2⋅3⋅a⋅b Express as prime factors

                     18bc = 2⋅3⋅3⋅b⋅c  

   The common factors  are 2⋅3⋅b.

        Therefore, the GCF is, 6b.

6ab + 18bc = (6b⋅a + 6b⋅3c) Take out the GCF, then divide the

polynomial using the GCFas

divisor to get the other factor.

                   =    = a + 3c The other factor of the given

polynomial 6ab + 18bc.  

                   =  6b (a + 3c) The factored form of 6ab + 18bc  

​Note:

The resulting expression is in factored form because it is written as a product of two polynomials, whereas the original expression is a two-termed sum.

Here are the steps in factoring polynomials with GCMF:

​1.  Find the greatest common monomial factor (GCMF). The largest monomial that is a factor of each term of the polynomial

2.  Factor it out, then divide the polynomial by the factor found in step 1. The quotient is the other factor.

3.  Express the polynomial as the product of two factors (the GCF and the quotient).

Remember:

The distributive property of multiplication over addition

𝒂 (𝒃 + 𝒄) = 𝒂𝒃 + 𝒂𝒄

In factoring out the greatest common factor we do its reverse.

Multiple Choice

Find all the prime factors of 30xy

• 2, 3, 6, x, y

• 2, 3, 6, x

• 2, 3, 5, y

• 2, 3, 5, x, y

Multiple Choice

Find all the prime factors of 42ab²

• 2, 3, 6, a, b

• 2, 3, 7, a, b

• 2, 3, 6, a, a, b

• 2, 3, 7, a, a, b

Fill in the Blank

Find the greatest common factor of 24y and 30xy

Multiple Choice

Find the greatest common factor of 6x³, 24x² and 8x

• 2x

• 3x

• 4x

• 6x

Multiple Choice

Factor 12x + 8y

• 2 (x + 4y)

• 4(3x + 2y)

• 8(x + y)

• 12(2x + 3y)

Fill in the Blank

Fill in the blank.

The _________ is the largest monomial that is a factor of each term of the polynomial.

Fill in the Blank

Fill in the blank.

Factor it out then, _______ the polynomial by the factor found in step 1.

The ________ is the other factor.

Fill in the Blank

Fill in the blank.

Express the polynomial as the ________ of two factors (the GCMF and the quotient).

DIFFERENCE OF TWO SQUARES (DOTS)

​SQUARE THE FOLLOWING NUMBERS:

Fill in the Blank

2² = ____

Fill in the Blank

5² = ____

Fill in the Blank

9² = ____

Fill in the Blank

16² = ____

Find the principal root of the numbers:

Fill in the Blank

  $$\sqrt[]{4}$$  = ____

Fill in the Blank

  $$\sqrt[]{25}$$  = ____

Fill in the Blank

  $$\sqrt[]{81}$$  = ____

Fill in the Blank

  $$\sqrt[]{256}$$  = ____

Simplify the following:

Multiple Choice

5² – 2² = _______

• 4

• 21

• 25

Multiple Choice

16² – 9² = _______

• 144

• 169

• 175

Fill in the Blank

(9 + 2) (9 – 2) = _______

Fill in the Blank

(16 + 5)(16 – 5) = ______

​Marikina City is known to be the shoe capital of the Philippines. Have you ever visited any shoe shops in this city? Have you observed how the shoes are arranged inside the stores? Why do you think they are arranged that way?

​In support to the local sapateros of the City of Marikina, the Marikina Cultural, Tourism, Trade and Investment Promotion Office will exhibit the shoe products of the 300 registered shoe and leather manufacturers in the city. The proposed alloted space for the exhibit is the freedom park. Organizers will install a façade inside the parameter of the said park and shall be divided equally for the participants.

​The product of the sum and difference of two polynomials is unique in the sense that its middle term vanishes. Since factoring is the reverse of finding the product, the difference of two squares is therefore, the product of the sum and difference of the square roots.

​

​​That is, x² – y² = (x + y)(x– y)

​

How do we factor the difference of two squares of polynomials?

Here are the steps in factoring polynomials as DOTS:

●       Find the greatest common monomial factor (GCMF), if any.

●       Express each term using the pattern, x² – y² = (x + y)(x– y)

    Which is the sum and difference of the square roots of the first and the

    last terms.

●       Check the results.​

​

​Example 1: Factor x² – 36y²

(1)  The polynomial x² – 36y² is obviously the difference of two squares

                                                                 without common factor.

(2)  Therefore, x² – 36y² = x² – 36y²

                      = (x)² - (6y)²

                    = (x + 6y)(x - 6y).

Factor the following polynomials completely.

Multiple Choice

a² – 4

• (a - 2) (a - 2)

• (a + 2) (a - 2)

• (a + 2) (a + 2)

Multiple Choice

25s² – t²

• (5s + t) (5s - t)

• (5 + t) (5 - t)

• (5s - 1) (5s +2)

Multiple Choice

9x² – 16y²

• (3x - 4y) (3x + 4y)

• (3x + 4y) (3x + 4y)

• (3x - y) (3x + y)

Open Ended

How did you factor the difference of two squares?

 

Write the steps:   ____________________________

                            ____________________________

                            ____________________________

​

​

Multiple Choice

Find the greatest common factor of 24, 64, 108.

• 2

• 4

• 8

• 24

Multiple Choice

What is the greatest common factor of the monomials a³b³ and a²b⁵?

• a²b³

• a²b⁵

• a³b³

• a³b⁵

Multiple Choice

What is the greatest common factor of the polynomial 30w³ + 48w + 12w²?

• 6w

• 6w³    

• 12w

• 12w³

Multiple Choice

What is the factored form of 56a³ – 8a?

• 8a (7a³ – a)

• 8a² (35a² – a)

• 8a (7a² – 1)

• 8a² (56a³ – 8a)

Multiple Choice

Complete the factor of 7x²y⁵ + 56x²y⁴ = 7x²y⁴ ( ______ ).

• y + 8

• 7y + 8x

• y + 8x²

• x²y + 8

Multiple Choice

Multiply (25 – 7) (25 + 7).

• 625

• 576    

• 484

• 49

Multiple Choice

Which of the following polynomials can be factored using the difference of two squares formula?

• x² + 16

• 25 – y²

• x³ + 1

• x³ – 27

Multiple Choice

When factored, the expression 16x² – 25y² is equivalent to __.

• (4x – 5y) (4x + 5y)

• (8x – 5y) (8x + 5y)

• (4x – 5y) (4x - 5y)

• (8x – 5y) (8x - 5y)

Multiple Choice

Factor 16x² – 100.

• (2x – 5) (2x + 5)

• 4(2x – 10) (2x + 10)

• 4(2x – 5) (2x + 5)

• 4(4x – 5) (4x + 5)

Multiple Choice

Factor x⁴ – y⁴ completely.

• (x² – y²) (x² – y²)

• (x² – y²) (x² + y²)

• (x – y) (x + y)

  (x² + y²)

• (x – y) (x + y)

  (x² - y²)