college algebra

Module 3:  Comfort with factoring polynomials

 

Discussion Instructions:

Please review  the class discussion directions . They tell you how to prepare and reflect before responding to the prompt below. They also include detailed guidelines how you must respond to classmates and then reply to the responses you receive. Please pay close attention to the response directions.  To earn your discussion points, you must post, respond, and reply as directed.

Discussion writing prompt

1.

1. Using a scale of 1 – 5 with 1 being “very uncomfortable” and 5 being “very comfortable,” briefly describe your level of comfort with factoring polynomials.

2. Provide an explanation as to why you gave the rating above.

3. Why is it important to consider the product rule for exponents when multiplying polynomials? Be sure to provide at least one example in your response.

Response focus

1. In your response, in addition to the standard requirement of noting what you have in common and where you differ with classmates, offer a suggestion as to how to improve your classmates' comfort level with the topics we are studying this week.

Remember to review the  academic expectations  for your submission.

Submission reminders:

· Submit your initial discussion post by 11:59 pm ET on Wednesday.  Respond to two of your classmates' discussion posts, and reply to the responses you received by 11:59 pm ET on Sunday.

· Contribute a minimum of 200 words to the initial post.

· Follow established  netiquetteLinks to an external site. guidelines when participating in forums.

Reply

 Module 3:  Polynomials and Polynomial Functions II

Factoring

Factoring is the opposite of multiplying. To factor an expression means to write it as a product of other expressions.

A polynomial is a finite sum of terms in which all variables have whole number exponents and no variable appears in a denominator. A monomial is a polynomial with one term. A binomial is a polynomial with two terms. A trinomial is a polynomial with three terms.

The Greatest Common Factor ( GCF) is the product of the factors common to all terms in the polynomial.

Find the Greatest Common Factor:

To factor a monomial from a polynomial, we need to factor out the Greatest Common Factor from each term in the polynomial.

Examples:

A. 6x + 21, the GCF is 3, then the factors of the polynomial are: 6x + 21 = 3(2x + 7)

B. 𝑥3𝑦2, 𝑥𝑦4, 𝑥5𝑦6, the highest power of x that is common to all three terms is x (𝑥1). The highest power of y that is common to all three terms is 𝑦2. Thus, the GCF of the three terms is 𝑥𝑦2.

Factor a Monomial from a Polynomial:

Steps:

1. Determine the GCF of all terms in the polynomial.

2. Write each term as the product of the GCF and another factor. 3) Use the distributive product to factor out the GCF.

Example:

Factor 15𝑥4 − 5𝑥3 + 25𝑥2

1. The GCF is 5𝑥2

2. Each term as the product of the GCF and another product: 5𝑥2 *3𝑥2 − 5𝑥2*𝑥 + 5𝑥2*5

3. Factor out the GCF: 5𝑥2(3𝑥2 − 𝑥 + 5)

Factor a Common Binomial Factor

Involves factoring a binomial as the GCF.

Example:

 Factor 9x(2x-3) + 7(2x-3) GCF: (2x-3) Factoring out the GCF: (2x-3)(9x+7)

Factor by Grouping

Involves factoring a polynomial that contains four terms. To factor by grouping, remove common factors from group of terms.

Example: Factor 𝑥3 + 2𝑥 − 5𝑥2 − 10

Steps

1. Determine if all four terms have a common factor. If so, factor out the GCF from each term.

2. Arrange the four terms into two groups of two terms each. Each group of two terms must have a GCF.

3. Factor the GCF from each group of two terms.

4. If the two terms formed in step c have a GCF, factor it out.

Note: In this example, there are no factors common to all four terms. Factor x from the first two terms and -5 from the last two term: 𝑥3 +2𝑥−5𝑥2 −10 = 𝑥(𝑥2 +2) − 5(𝑥2 +2)  = (𝑥2 + 2)(𝑥−5)

Factoring Trinomials

Factor Trinomials of the Form 𝑥2 + 𝑏𝑥 + 𝑐

Find two numbers (or factors) whose product is c and whose sum is b.

Example:

Factor 𝑥2 − 𝑥 − 12

a= 1 b = -1 c = -12  

find two numbers whose product is -12 and whose sum is -1 (x + 3) (x – 4)

Factor Trinomials of the Form 𝑎𝑥2 + 𝑏𝑥 + 𝑐

1. Find two numbers whose product is a.c and whose sum is b

2. Rewrite the middle term, bx, using the numbers found in step 1

3. Factor by grouping.

Example: Factor 2𝑥2 − 5𝑥 − 12

a = 2 b = -5 c = -12

Steps:

1. Find two numbers whose product is a.c and whose sum is b:  2(-12) = 24, sum = -5 The two numbers are -8 and 3

2. Rewrite the middle term, bx, using the numbers found in step 1: -5x, using -8x and 3x:  -8x + 3x

3. Factor by grouping:

A. 2𝑥2 − 8𝑥 + 3𝑥 − 12. 

B. That gets: 2x(x -4) + 3 (x – 4)

C. (x – 4) (2x + 3) 

Special Factoring Formulas

Factoring the difference of two squares:

𝑎2 − 𝑏2 = (𝑎 + 𝑏)(𝑎 − 𝑏)

Example:

Factor 𝑥2 − 16

   𝑥2−16 = (𝑥)2−(4)2 = (𝑥+4)(𝑥−4)

Quadratic Equations

The formula −b±b2−4ac2a can be used to solve quadratic equations in the form of: 𝑎𝑥2 + 𝑏𝑥 + 𝑐

Example:

Solve 𝑥2 + 2𝑥 − 8

a= 1 b= 2 c = -8

𝑥 = −b±b2−4ac2a  = −2±22−4(1)(−8)2(1) = −2±4+322

                

 =   −2±362 = −2±62 = 𝑥 = −2−62orx=−2+62

x=−4orx=2                

References

Angel, A., & Rundle, D. (2018).  Intermediate algebra for college students. (10th edition). Pearson.