Calculator (college algebra): Factoring using the TI 84

Susan Jones
3 Mar 201109:30
EducationalLearning
32 Likes 10 Comments

TLDRThe video explains how to factor quadratic polynomials using a graphing calculator. It demonstrates this with three examples: \(x^2 + 13x + 40\), \(x^2 - 5x - 14\), and \(6x^2 + 19x - 7\). For each polynomial, the video shows how to find the x-intercepts on the graph, which correspond to the factors of the polynomial. For the last example, it details converting decimal intercepts to fractional form and adjusting the factors to eliminate fractions, ensuring the product matches the original polynomial.

Takeaways
  • 🔍 The video script explains how to factor quadratic polynomials using a calculator.
  • 📈 The process involves identifying the x-intercepts of the polynomial graph, which are the points where the graph crosses the x-axis.
  • 📝 For the polynomial \( x^2 + 13x + 40 \), the factors are found to be \( (x + 8) \) and \( (x - 5) \) by observing the x-intercepts at -5 and 8.
  • 🔢 The script demonstrates using the calculator's graphing and calculation functions to verify the x-intercepts and to factor the polynomials.
  • 📉 When graphing, adjusting the viewing window can help in identifying the x-intercepts more clearly.
  • 🤔 The script acknowledges that sometimes x-intercepts may not be integers, requiring the use of decimal or fraction values for the factors.
  • 📝 For the polynomial \( 6x^2 + 19x - 7 \), the factors are determined to be \( 2x + 7 \) and \( 3x - \frac{1}{3} \) after converting the decimal x-intercepts to fractions.
  • 🔄 The script emphasizes the importance of multiplying the factors back together to verify the correctness of the factorization.
  • 📚 It is mentioned that a calculator can simplify the process of finding x-intercepts and factoring polynomials.
  • 🛠️ The script provides a step-by-step guide on how to use the calculator to assist in the factorization process.
  • 📉 The video also covers how to handle polynomials where the x-intercepts are not immediately obvious, requiring more precise calculations.
Q & A

  • What is the purpose of the video script?

    -The purpose of the video script is to demonstrate how to factor quadratic polynomials using a calculator and the process of finding x-intercepts to determine the factors.

  • What is the first quadratic polynomial discussed in the script?

    -The first quadratic polynomial discussed is x^2 + 13x + 40.

  • How does the script suggest finding the factors of the first polynomial?

    -The script suggests using the calculator to graph the polynomial and identify the x-intercepts, which are then used to determine the factors.

  • What are the x-intercepts of the first polynomial found in the script?

    -The x-intercepts of the first polynomial are -5 and 8.

  • How does the script verify the factors of the first polynomial?

    -The script verifies the factors by multiplying the binomials (x + 8) and (x + 5) back together to ensure they yield the original polynomial.

  • What is the second polynomial discussed in the script?

    -The second polynomial discussed is x^2 - 5x - 14.

  • How does the script identify the x-intercepts for the second polynomial?

    -The script identifies the x-intercepts by graphing the polynomial and observing where the graph crosses the x-axis.

  • What are the x-intercepts of the second polynomial found in the script?

    -The x-intercepts of the second polynomial are -2 and 7.

  • What is the third polynomial discussed in the script?

    -The third polynomial discussed is 6x^2 + 19x - 7.

  • How does the script handle the non-integer x-intercepts of the third polynomial?

    -The script uses the calculator to find the non-integer x-intercepts and then converts them into factors by multiplying the binomials to eliminate fractions.

  • What are the factors of the third polynomial found in the script?

    -The factors of the third polynomial are 2x + 7 and 3x - 1/3, which are derived from the x-intercepts -3.5 and 1/3.

  • How does the script ensure the accuracy of the factors for the third polynomial?

    -The script ensures accuracy by multiplying the factors back together to check if they result in the original polynomial.

Outlines
00:00
📚 Factoring Quadratic Polynomials Using Graphs

This paragraph introduces the process of factoring quadratic polynomials with the help of a calculator. The speaker demonstrates how to identify factors by graphing the polynomial and finding the x-intercepts. The example given is 'x squared plus 13x plus 40', which is factored into '(x + 8)(x + 5)' after verifying the x-intercepts at -5 and 8. The speaker also emphasizes the importance of verifying the factorization by multiplying the binomials back together.

05:02
📉 Graphing to Find Non-Integer X-Intercepts

The second paragraph continues the discussion on factoring polynomials but focuses on non-integer x-intercepts. The example '6x squared plus 19x minus 7' is used to illustrate how to adjust the graph's viewing window to find the x-intercepts. The speaker explains the process of calculating the x-intercepts, which are found to be -3.5 and 1/3. The challenge of converting these non-integer intercepts into factors is addressed, and the solution involves multiplying the factors by appropriate numbers to eliminate fractions and achieve the original polynomial form.

Mindmap
Keywords
💡Factor
In the context of the video, 'factor' refers to the process of breaking down a polynomial into a product of simpler polynomials, typically binomials. This is a fundamental concept in algebra and is essential for solving equations and understanding the behavior of polynomial functions. The script discusses factoring quadratic polynomials, such as 'x squared plus 13x plus 40', into two binomials that multiply to give the original expression.
💡Quadratic Polynomials
A 'quadratic polynomial' is a polynomial of degree two, having the general form 'ax^2 + bx + c', where 'a', 'b', and 'c' are constants, and 'a' is not zero. The video focuses on factoring specific quadratic polynomials, demonstrating how to identify and use their x-intercepts to find the factors.
💡Calculator
The 'calculator' in the script is used as a tool to assist in the process of factoring polynomials. It helps in graphing the polynomial to find its x-intercepts, which are crucial for determining the factors. The script mentions using the calculator's graphing and calculation functions to verify the x-intercepts and to simplify the factoring process.
💡Graph
A 'graph' in the script refers to the visual representation of a polynomial function, typically plotted on a coordinate plane. The graph is used to identify the x-intercepts, which are the points where the graph crosses the x-axis. These points are essential for factoring the polynomial, as demonstrated in the script.
💡X-Intercept
An 'x-intercept' is the point where a graph of a function crosses the x-axis. In the context of the video, finding the x-intercepts is a step in the factoring process, as they correspond to the roots of the polynomial. The script shows how to use the calculator to find these intercepts for polynomials like 'x squared plus 13x plus 40'.
💡Binomials
'Binomials' are algebraic expressions with two terms, such as 'x + 8' or '2x - 7'. In the video, the factoring process involves expressing a quadratic polynomial as a product of two binomials, which is a common method for solving quadratic equations.
💡Polynomial
A 'polynomial' is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, and multiplication, and non-negative integer exponents of variables. The script deals with factoring specific types of polynomials, namely quadratic polynomials.
💡Standard Viewing Window
The 'standard viewing window' refers to the default settings for the range of the x and y axes when graphing functions on a calculator. The script mentions adjusting this window to better visualize the graph of the polynomial and identify the x-intercepts.
💡Roots
'Roots' of a polynomial are the values of the variable for which the polynomial equals zero. In the script, finding the roots is synonymous with identifying the x-intercepts, which are then used to determine the factors of the polynomial.
💡Multiplying Binomials
The process of 'multiplying binomials' involves using the distributive property to expand the product of two binomials into a polynomial. In the script, after identifying potential factors, the presenter multiplies them to verify that they indeed produce the original polynomial.
💡Fraction
A 'fraction' is a numerical expression that represents division, consisting of a numerator and a denominator. In the script, when dealing with non-integer x-intercepts, the presenter converts decimal values to fractions to find exact factors for the polynomial, as seen with the x-intercept of '-3.5' being converted to '-7/2'.
Highlights

The calculator can assist in factoring polynomials, specifically quadratic ones.

Quadratic polynomials are factored into two binomials with specific properties.

The product of the first terms should equal the quadratic term, and the product of the last terms should equal the constant term.

The sum of the products of the binomials should match the linear term of the polynomial.

The graph of the polynomial can help identify the x-intercepts, which are crucial for factoring.

Calculator can verify the x-intercepts by calculating the value of y when x is at the suspected intercept.

The x-intercepts can be used to form the factors of the polynomial.

Multiplying the factors back together can verify the correctness of the factoring.

The process of factoring can be demonstrated using the example of x squared minus 5x minus 14.

The x-intercepts for the polynomial x squared minus 5x minus 14 are identified as -2 and 7.

The factors of the polynomial are verified by multiplying them back together.

The process can become more complex with polynomials that do not have clear integer x-intercepts.

Adjusting the viewing window on the calculator can help identify non-integer x-intercepts.

Non-integer x-intercepts can be found using the calculator's calculate function.

When dealing with non-integer intercepts, it's important to express them in fraction form to avoid rounding errors.

The process of factoring 6x squared plus 19x minus 7 involves dealing with decimal x-intercepts and converting them into fractions.

The factors of the polynomial are verified by multiplying them and ensuring they match the original polynomial.

Transcripts

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