Find The Equation Having The Given Roots:i) − 1 3 -\frac{1}{3} − 3 1 ​ And 1 2 \frac{1}{2} 2 1 ​ Ii) 12 3 \frac{12}{3} 3 12 ​ And − 12 3 -\frac{12}{3} − 3 12 ​

Introduction

In algebra, finding the equation of a polynomial given its roots is a fundamental concept. This concept is crucial in solving various mathematical problems, particularly in the field of calculus and engineering. In this article, we will explore how to find the equation of a polynomial given its roots.

Finding the Equation of a Polynomial Given Its Roots

To find the equation of a polynomial given its roots, we can use the factored form of a polynomial. The factored form of a polynomial is given by:

$$a(x - r_1)(x - r_2)...(x - r_n)$$

where $a$ is the leading coefficient, and $r_1, r_2,...,r_n$ are the roots of the polynomial.

Case i: $-\frac{1}{3}$ and $\frac{1}{2}$

Let's consider the case where the roots are $-\frac{1}{3}$ and $\frac{1}{2}$. We can write the factored form of the polynomial as:

$$a(x + \frac{1}{3})(x - \frac{1}{2})$$

To find the value of $a$, we can use the fact that the product of the roots is equal to the constant term of the polynomial divided by the leading coefficient. In this case, the product of the roots is:

$$(-\frac{1}{3})(\frac{1}{2}) = -\frac{1}{6}$$

Since the constant term of the polynomial is $-\frac{1}{6}$, we can set up the equation:

$$a(-\frac{1}{6}) = -\frac{1}{6}$$

Solving for $a$, we get:

$$a = 1$$

Therefore, the equation of the polynomial is:

$$x^2 + \frac{1}{3}x - \frac{1}{2}$$

Case ii: $\frac{12}{3}$ and $-\frac{12}{3}$

Let's consider the case where the roots are $\frac{12}{3}$ and $-\frac{12}{3}$. We can write the factored form of the polynomial as:

$$a(x - \frac{12}{3})(x + \frac{12}{3})$$

To find the value of $a$, we can use the fact that the product of the roots is equal to the constant term of the polynomial divided by the leading coefficient. In this case, the product of the roots is:

$$(\frac{12}{3})(-\frac{12}{3}) = -\frac{144}{9} = -16$$

Since the constant term of the polynomial is $-16$, we can set up the equation:

$$a(-16) = -16$$

Solving for $a$, we get:

$$a = 1$$

Therefore, the equation of the polynomial is:

$$x^2 - 16$$

Conclusion

In conclusion, finding the equation of a polynomial given its roots is a straightforward process that involves using the factored form of a polynomial. By following the steps outlined in this article, we can easily find the equation of a polynomial given its roots.

Final Thoughts

Finding the equation of a polynomial given its roots is an essential concept in algebra that has numerous applications in various fields, including calculus and engineering. By mastering this concept, we can solve a wide range of mathematical problems and gain a deeper understanding of the underlying mathematics.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Engineering Mathematics" by John Bird

Further Reading

• [1] "Polynomial Equations" by Wolfram MathWorld
• [2] "Roots of a Polynomial" by Math Open Reference
• [3] "Factored Form of a Polynomial" by Purplemath
  

Introduction

In our previous article, we explored how to find the equation of a polynomial given its roots. In this article, we will answer some frequently asked questions related to finding the equation of a polynomial given its roots.

Q&A

Q: What is the factored form of a polynomial?

A: The factored form of a polynomial is given by:

$$a(x - r_1)(x - r_2)...(x - r_n)$$

where $a$ is the leading coefficient, and $r_1, r_2,...,r_n$ are the roots of the polynomial.

Q: How do I find the value of $a$ in the factored form of a polynomial?

A: To find the value of $a$, you can use the fact that the product of the roots is equal to the constant term of the polynomial divided by the leading coefficient.

Q: What is the product of the roots of a polynomial?

A: The product of the roots of a polynomial is equal to the constant term of the polynomial divided by the leading coefficient.

Q: How do I find the equation of a polynomial given its roots?

A: To find the equation of a polynomial given its roots, you can use the factored form of a polynomial and multiply the factors together.

Q: What is the difference between the factored form and the standard form of a polynomial?

A: The factored form of a polynomial is a product of factors, while the standard form of a polynomial is a sum of terms.

Q: Can I have multiple roots for a polynomial?

A: Yes, a polynomial can have multiple roots. In this case, the factored form of the polynomial will have multiple factors.

Q: How do I find the roots of a polynomial given its equation?

A: To find the roots of a polynomial given its equation, you can use various methods such as factoring, the quadratic formula, or numerical methods.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that gives the roots of a quadratic polynomial in the form of:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

where $a$, $b$, and $c$ are the coefficients of the quadratic polynomial.

Q: Can I use the quadratic formula to find the roots of a polynomial with multiple roots?

A: No, the quadratic formula is only applicable to quadratic polynomials with two roots. For polynomials with multiple roots, you will need to use other methods such as factoring or numerical methods.

Conclusion

In conclusion, finding the equation of a polynomial given its roots is a fundamental concept in algebra that has numerous applications in various fields. By mastering this concept, we can solve a wide range of mathematical problems and gain a deeper understanding of the underlying mathematics.

Final Thoughts

Finding the equation of a polynomial given its roots is an essential concept in algebra that has numerous applications in various fields, including calculus and engineering. By mastering this concept, we can solve a wide range of mathematical problems and gain a deeper understanding of the underlying mathematics.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Engineering Mathematics" by John Bird

Further Reading

• [1] "Polynomial Equations" by Wolfram MathWorld
• [2] "Roots of a Polynomial" by Math Open Reference
• [3] "Factored Form of a Polynomial" by Purplemath