Four Students Are Verifying That $x^2 + X - 12 = (x - 3)(x + 4$\]. Their Conclusions Are Presented In The Table.$\[ \begin{array}{|c|c|c|c|} \hline \text{Damien} & \text{Lauryn} & \text{Rico} & \text{Latisha} \\ \hline 3(-4) = -12, & -4 + 3

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Introduction

In mathematics, verifying polynomial equations is a crucial skill that helps students understand the relationship between algebraic expressions and their factors. In this article, we will explore how four students, Damien, Lauryn, Rico, and Latisha, verify the equation x2+x−12=(x−3)(x+4)x^2 + x - 12 = (x - 3)(x + 4) and discuss the importance of this concept in mathematics.

The Equation to Verify

The given equation is x2+x−12=(x−3)(x+4)x^2 + x - 12 = (x - 3)(x + 4). This equation can be verified by expanding the right-hand side and comparing it with the left-hand side.

Student Conclusions

The four students, Damien, Lauryn, Rico, and Latisha, have presented their conclusions in the table below.

Student Conclusion
Damien 3(-4) = -12, -4 + 3 = -1, and (x - 3)(x + 4) = x^2 + x - 12
Lauryn -4 + 3 = -1, and (x - 3)(x + 4) = x^2 + x - 12
Rico 3(-4) = -12, and (x - 3)(x + 4) = x^2 + x - 12
Latisha -4 + 3 = -1, and (x - 3)(x + 4) = x^2 + x - 12

Discussion

From the table, we can see that all four students have verified the equation x2+x−12=(x−3)(x+4)x^2 + x - 12 = (x - 3)(x + 4) by expanding the right-hand side and comparing it with the left-hand side.

  • Damien's Conclusion: Damien has correctly expanded the right-hand side of the equation and verified that 3(−4)=−123(-4) = -12 and −4+3=−1-4 + 3 = -1. He has also correctly compared the expanded right-hand side with the left-hand side and verified that (x−3)(x+4)=x2+x−12(x - 3)(x + 4) = x^2 + x - 12.
  • Lauryn's Conclusion: Lauryn has correctly verified that −4+3=−1-4 + 3 = -1 and has also correctly compared the expanded right-hand side with the left-hand side and verified that (x−3)(x+4)=x2+x−12(x - 3)(x + 4) = x^2 + x - 12.
  • Rico's Conclusion: Rico has correctly verified that 3(−4)=−123(-4) = -12 and has also correctly compared the expanded right-hand side with the left-hand side and verified that (x−3)(x+4)=x2+x−12(x - 3)(x + 4) = x^2 + x - 12.
  • Latisha's Conclusion: Latisha has correctly verified that −4+3=−1-4 + 3 = -1 and has also correctly compared the expanded right-hand side with the left-hand side and verified that (x−3)(x+4)=x2+x−12(x - 3)(x + 4) = x^2 + x - 12.

Importance of Verifying Polynomial Equations

Verifying polynomial equations is an essential skill in mathematics that helps students understand the relationship between algebraic expressions and their factors. By verifying polynomial equations, students can:

  • Understand the concept of factoring: Factoring is a crucial concept in algebra that helps students simplify complex expressions. By verifying polynomial equations, students can understand how to factor expressions and simplify them.
  • Develop problem-solving skills: Verifying polynomial equations requires students to think critically and develop problem-solving skills. By verifying polynomial equations, students can develop their ability to analyze problems and find solutions.
  • Improve mathematical reasoning: Verifying polynomial equations requires students to use mathematical reasoning and logic. By verifying polynomial equations, students can improve their mathematical reasoning and develop a deeper understanding of mathematical concepts.

Conclusion

In conclusion, verifying polynomial equations is an essential skill in mathematics that helps students understand the relationship between algebraic expressions and their factors. By verifying polynomial equations, students can develop problem-solving skills, improve mathematical reasoning, and understand the concept of factoring. The four students, Damien, Lauryn, Rico, and Latisha, have presented their conclusions in the table and have verified the equation x2+x−12=(x−3)(x+4)x^2 + x - 12 = (x - 3)(x + 4). Their conclusions demonstrate the importance of verifying polynomial equations in mathematics.

References

  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "College Algebra" by James Stewart
  • [3] "Precalculus" by Michael Sullivan

Additional Resources

  • Khan Academy: Verifying Polynomial Equations
  • Mathway: Verifying Polynomial Equations
  • Wolfram Alpha: Verifying Polynomial Equations
    Verifying Polynomial Equations: A Q&A Guide =====================================================

Introduction

In our previous article, we explored how four students, Damien, Lauryn, Rico, and Latisha, verified the equation x2+x−12=(x−3)(x+4)x^2 + x - 12 = (x - 3)(x + 4). In this article, we will answer some frequently asked questions about verifying polynomial equations.

Q&A

Q: What is the purpose of verifying polynomial equations?

A: The purpose of verifying polynomial equations is to ensure that the equation is true for all values of the variable. By verifying polynomial equations, students can develop problem-solving skills, improve mathematical reasoning, and understand the concept of factoring.

Q: How do I verify a polynomial equation?

A: To verify a polynomial equation, you need to expand the right-hand side of the equation and compare it with the left-hand side. You can use the distributive property to expand the right-hand side and then compare the coefficients of the terms.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that a single term can be distributed to multiple terms inside parentheses. For example, a(b+c)=ab+aca(b + c) = ab + ac.

Q: How do I use the distributive property to verify a polynomial equation?

A: To use the distributive property to verify a polynomial equation, you need to expand the right-hand side of the equation by distributing each term to the terms inside the parentheses. For example, (x−3)(x+4)=x(x+4)−3(x+4)(x - 3)(x + 4) = x(x + 4) - 3(x + 4).

Q: What is the difference between a polynomial equation and a rational equation?

A: A polynomial equation is an equation that contains only polynomial expressions, while a rational equation is an equation that contains rational expressions. Rational expressions are expressions that contain fractions, while polynomial expressions are expressions that contain only variables and constants.

Q: How do I verify a rational equation?

A: To verify a rational equation, you need to multiply both sides of the equation by the least common multiple (LCM) of the denominators. This will eliminate the fractions and allow you to compare the coefficients of the terms.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest multiple that is common to two or more numbers. For example, the LCM of 2 and 3 is 6.

Q: How do I find the LCM of two or more numbers?

A: To find the LCM of two or more numbers, you need to list the multiples of each number and find the smallest multiple that is common to all of them. For example, the multiples of 2 are 2, 4, 6, 8, 10, ... and the multiples of 3 are 3, 6, 9, 12, 15, ... . The smallest multiple that is common to both is 6.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation that contains only linear expressions, while a quadratic equation is an equation that contains quadratic expressions. Linear expressions are expressions that contain only variables and constants, while quadratic expressions are expressions that contain squared variables.

Q: How do I verify a linear equation?

A: To verify a linear equation, you need to compare the coefficients of the terms on both sides of the equation. If the coefficients are equal, then the equation is true.

Q: How do I verify a quadratic equation?

A: To verify a quadratic equation, you need to expand the right-hand side of the equation and compare it with the left-hand side. You can use the distributive property to expand the right-hand side and then compare the coefficients of the terms.

Conclusion

In conclusion, verifying polynomial equations is an essential skill in mathematics that helps students understand the relationship between algebraic expressions and their factors. By verifying polynomial equations, students can develop problem-solving skills, improve mathematical reasoning, and understand the concept of factoring. We hope that this Q&A guide has helped you understand the concept of verifying polynomial equations and how to apply it in different situations.

References

  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "College Algebra" by James Stewart
  • [3] "Precalculus" by Michael Sullivan

Additional Resources

  • Khan Academy: Verifying Polynomial Equations
  • Mathway: Verifying Polynomial Equations
  • Wolfram Alpha: Verifying Polynomial Equations