How To Factor Polynomials The Easy Way!

The Organic Chemistry Tutor
18 Nov 201911:53
EducationalLearning
32 Likes 10 Comments

TLDRThis instructional video offers a comprehensive guide on factoring polynomials with various examples. It begins with extracting the greatest common factor (GCF) from simple polynomial expressions and progresses to factoring trinomials with a leading coefficient of one, then moves on to more complex scenarios, including expressions with higher degree terms and non-GCF scenarios. The video also covers factoring by grouping and handling perfect square differences. It concludes with strategies for factoring polynomials with four terms and similar ratios, providing a clear and structured approach to mastering polynomial factoring.

Takeaways
  • 📚 Start by identifying the greatest common factor (GCF) in polynomial expressions to simplify them.
  • 🔍 For expressions like 3x^3 - 9x^2 and 4x^2 - 12x, factor out the GCF and simplify the terms to get x(x^2 - 3x) and 4x(x - 3) respectively.
  • 📈 Factoring trinomials with a leading coefficient of one involves finding two numbers that multiply to the constant term and add to the middle coefficient.
  • 🔑 Example: x^2 + 2x - 15 factors to (x + 5)(x - 3) by using the numbers 5 and -3.
  • 🌐 When the leading coefficient is not one, like in 2x^2 - 6x - 56, first factor out the GCF and then proceed to factor the resulting trinomial.
  • 📉 For expressions like 3x^2 - 18x + 24, factor out the GCF and then find two numbers that multiply to the constant term and add to the middle coefficient to complete the factorization.
  • 💡 Factoring expressions like x^2 - 16 or x^2 - 64 involves recognizing and applying the difference of perfect squares formula.
  • 📌 The square root of the squared term becomes the base of the binomial factors, and the square root of the constant term determines the signs of the binomial factors.
  • 🔄 For expressions that don't have an obvious GCF, like 2x^2 - 5x - 3, multiply the leading coefficient by the constant term to find two numbers that add to the middle coefficient and multiply to the product.
  • 📝 Factor by grouping, which involves taking out common factors from groups of terms and then factoring out the common binomial from the resulting expression.
  • 🧩 Practice factoring by working through examples like 6x^2 + x - 15, which requires a combination of the methods discussed in the video.
Q & A

  • What is the first step in factoring the polynomial 6x - 12?

    -The first step is to take out the greatest common factor (GCF), which is 6, resulting in 6(x - 2).

  • How is the polynomial 3x^3 - 9x^2 factored?

    -The GCF of 3x^2 is taken out, resulting in x^2(3x - 9), and then further simplified to x^2(3)(x - 3).

  • What is the process for factoring the expression 4x^2 - 12x?

    -First, take out the GCF which is 4x, resulting in x(4x - 12), and then simplify to x(2)(x - 3).

  • When factoring trinomials with a leading coefficient of one, what should you look for?

    -You should look for two numbers that multiply to the constant term and add up to the linear coefficient.

  • How do you factor the trinomial x^2 + 2x - 15?

    -You find two numbers that multiply to -15 and add up to 2, which are 5 and -3, resulting in (x + 5)(x - 3).

  • What is the first step in factoring the polynomial 2x^2 - 6x - 56?

    -The first step is to factor out the GCF, which is 2, resulting in 2(x^2 - 3x - 28).

  • How is the expression 3x^2 - 18x + 24 factored?

    -First, factor out the GCF which is 3, resulting in 3(x^2 - 6x + 8), and then factor the trinomial inside to get 3(x - 4)(x - 2).

  • What is a difference of perfect squares and how is it factored?

    -A difference of perfect squares is an expression of the form a^2 - b^2, and it is factored as (a + b)(a - b).

  • How do you approach factoring the polynomial 2x^2 - 5x - 3?

    -Multiply the leading coefficient by the constant term to get -6, then find two numbers that multiply to -6 and add to -5, which are -6 and 1, and use factoring by grouping.

  • What is the process for factoring the polynomial 3x^3 - 2x^2 - 12x + 8?

    -First, notice the ratio between the coefficients and factor by grouping, taking out x^2 and -4 from the respective terms, resulting in (x^2 - 4)(3x - 2), and then factor the difference of squares to get (x + 2)(x - 2)(3x - 2).

Outlines
00:00
📚 Introduction to Factoring Polynomials

This paragraph introduces the concept of factoring polynomials with simple examples. It explains how to factor expressions like 6x - 12 by taking out the greatest common factor (GCF), which is 6, resulting in x - 2. The process is repeated for 3x^3 - 9x^2 and 4x^2 - 12x, where the GCFs are x^2 and 4x, respectively. The paragraph then transitions to discussing the factoring of trinomials with a leading coefficient of one, using the example of x^2 + 7x + 12, which factors into (x + 3)(x + 4). The method involves finding two numbers that multiply to the constant term and add up to the linear coefficient.

05:00
🔍 Factoring Trinomials and Special Cases

This paragraph delves deeper into factoring trinomials, particularly when the leading coefficient is not one. It provides a step-by-step example of how to factor x^2 + 2x - 15 by identifying two numbers that multiply to -15 and add up to 2, which are 5 and -3, leading to the factored form (x + 5)(x - 3). The paragraph also covers the factoring of expressions with a leading coefficient other than one, such as 2x^2 - 6x - 56, by first factoring out the GCF of 2 and then applying the same trinomial factoring technique. It also touches on the factoring of perfect square trinomials like x^2 - 64 and the difference of squares, demonstrating the process with examples.

10:01
📘 Advanced Factoring Techniques

The final paragraph discusses advanced factoring techniques, including factoring by grouping and handling polynomials with more than three terms. It provides an example of factoring a four-term polynomial, 3x^3 - 2x^2 - 12x + 8, by recognizing a common ratio in the coefficients and factoring by grouping. The paragraph also explains how to deal with trinomials that do not have a common factor and cannot be factored by simple methods, such as 2x^2 - 5x - 3, by using the 'ac method' and factoring by grouping. The video concludes with a summary of the factoring techniques covered and an invitation to explore more examples through the provided links.

Mindmap
Keywords
💡Factoring Polynomials
Factoring polynomials is the process of expressing a polynomial as the product of its factors, which are simpler polynomials or expressions. This is central to the video's theme as it demonstrates various methods to break down complex polynomials into their constituent parts. For example, the script starts by factoring '6x - 12' by taking out the greatest common factor (GCF), which is 6.
💡Greatest Common Factor (GCF)
The GCF is the largest number that divides two or more numbers without leaving a remainder. In the context of the video, it is used to simplify polynomials by factoring out the common factor from all terms. The script illustrates this with the polynomial '4x^2 - 12x', where 4x is factored out to simplify the expression.
💡Trinomial
A trinomial is a polynomial with three terms. The video discusses factoring trinomials, especially when the leading coefficient is one, by finding two numbers that multiply to the constant term and add up to the linear coefficient. This is exemplified in the script with 'x^2 + 2x - 15', which factors into '(x + 5)(x - 3)'.
💡Difference of Perfect Squares
This is a specific type of polynomial that can be factored into the product of two binomials. It is characterized by the expression 'a^2 - b^2', which factors into '(a + b)(a - b)'. The video mentions this concept when discussing the factoring of 'x^2 - 16' and 'x^2 - 64', which are both differences of perfect squares.
💡Factoring by Grouping
Factoring by grouping is a technique used when a polynomial has more than three terms and the terms do not share a common factor. The video describes this method by first multiplying the leading coefficient by the constant term and then finding two numbers that meet specific multiplication and addition criteria. The script applies this to '2x^2 - 5x - 3', which is factored into '2(x - 3)(x + 1)'.
💡Leading Coefficient
The leading coefficient is the coefficient of the term with the highest power in a polynomial. The video emphasizes the importance of this term in the factoring process, especially when it is not one, as it affects the approach to factoring. For instance, in '2x^2 - 6x - 56', the leading coefficient is 2, and the script shows how factoring out this coefficient simplifies the problem.
💡Polynomial Expression
A polynomial expression is a mathematical expression involving a sum of powers in a single variable. The video's content revolves around different types of polynomial expressions and their factoring techniques. The script provides several examples of polynomials, such as '3x^3 - 9x^2' and '4x^2 - 12x', and demonstrates how to factor them.
💡Perfect Square
A perfect square is a number or an expression that can be expressed as the square of an integer or another integer expression. In the context of the video, perfect squares are used to identify and factor expressions of the form 'a^2 - b^2'. The script explains how 'x^2 - 16' is a difference of perfect squares and factors into '(x + 4)(x - 4)'.
💡Binomial
A binomial is an algebraic expression with two terms. The video discusses binomials in the context of factoring trinomials and perfect squares. For example, when factoring 'x^2 - 64', the script shows that it can be expressed as the product of two binomials: '(x + 8)(x - 8)'.
💡Coefficient
In a polynomial, the coefficient is the numerical factor multiplying the variable. The video script refers to coefficients when discussing the factoring process, such as in '3x^3 - 18x + 24', where the coefficients 3, -18, and 24 are considered to factor out the GCF of 3.
Highlights

Introduction to factoring polynomials with simple examples.

Factoring 6x - 12 by taking out the greatest common factor (GCF).

Factoring 3x^3 - 9x^2 by identifying the GCF of 3x^2.

Factoring 4x^2 - 12x by taking out the GCF of 4x.

Explanation of factoring trinomials with a leading coefficient of one.

Factoring x^2 + 2x - 15 by finding numbers that multiply to -15 and add to 2.

Factoring 2x^2 - 6x - 56 after factoring out the GCF of 2.

Factoring 3x^2 - 18x + 24 by taking out the GCF of 3.

Factoring expressions like x^2 - 16 and x^2 - 64 as differences of perfect squares.

Factoring 2x^2 - 5x - 3 using the method of factoring by grouping.

Approach to factor 6x^2 + x - 15 by multiplying the leading coefficient by the constant term.

Factoring 3x^3 - 2x^2 - 12x + 8 using the ratio of coefficients for factoring by grouping.

Final example of factoring a four-term polynomial with a similar ratio in coefficients.

Factoring a difference of perfect squares situation in the final example.

Conclusion and thanks for watching with a prompt to check out more examples in the description.

Transcripts

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Polynomial FactoringMath EducationAlgebra TechniquesGreatest Common FactorTrinomial FactoringPerfect SquaresDifference of SquaresFactoring by GroupingMath TutorialEducational Content