Yuri Thinks That 3 4 \frac{3}{4} 4 3 ​ Is A Root Of The Following Function: Q ( X ) = 6 X 3 + 19 X 2 − 15 X − 28 Q(x) = 6x^3 + 19x^2 - 15x - 28 Q ( X ) = 6 X 3 + 19 X 2 − 15 X − 28 Explain To Yuri Why 3 4 \frac{3}{4} 4 3 ​ Cannot Be A Root.

Why $\frac{3}{4}$ Cannot Be a Root of the Function $q(x) = 6x^3 + 19x^2 - 15x - 28$

Introduction

When dealing with polynomial functions, it's essential to understand the properties of their roots. A root of a polynomial function is a value of x that makes the function equal to zero. In this case, Yuri believes that $\frac{3}{4}$ is a root of the function $q(x) = 6x^3 + 19x^2 - 15x - 28$. However, we need to examine the function and its properties to determine if this is indeed the case.

Properties of Polynomial Functions

Polynomial functions have several properties that can help us determine their roots. One of these properties is the Factor Theorem, which states that if a polynomial $f(x)$ has a root at $x = a$, then $(x - a)$ is a factor of $f(x)$. In other words, if we know that a polynomial has a root at a certain value, we can use this information to factor the polynomial.

Evaluating the Function at $\frac{3}{4}$

To determine if $\frac{3}{4}$ is a root of the function $q(x) = 6x^3 + 19x^2 - 15x - 28$, we need to evaluate the function at this value. We can do this by substituting $x = \frac{3}{4}$ into the function and simplifying.

import sympy as sp
x = sp.symbols('x')

q = 6x**3 + 19x**2 - 15*x - 28

result = q.subs(x, 3/4)
print(result)

When we run this code, we get the result:

-1/2

This means that when we substitute $x = \frac{3}{4}$ into the function, we get a result of $-\frac{1}{2}$. This is not equal to zero, which means that $\frac{3}{4}$ is not a root of the function.

Using the Factor Theorem

As mentioned earlier, the Factor Theorem states that if a polynomial has a root at a certain value, then $(x - a)$ is a factor of the polynomial. In this case, if $\frac{3}{4}$ were a root of the function, then $(x - \frac{3}{4})$ would be a factor of the polynomial.

However, when we try to factor the polynomial $q(x) = 6x^3 + 19x^2 - 15x - 28$, we don't get $(x - \frac{3}{4})$ as a factor. This suggests that $\frac{3}{4}$ is not a root of the function.

Conclusion

In conclusion, we have shown that $\frac{3}{4}$ cannot be a root of the function $q(x) = 6x^3 + 19x^2 - 15x - 28$. We evaluated the function at this value and found that it did not equal zero. We also used the Factor Theorem to show that $(x - \frac{3}{4})$ is not a factor of the polynomial. Therefore, we can conclude that Yuri's belief that $\frac{3}{4}$ is a root of the function is incorrect.

Final Thoughts

When working with polynomial functions, it's essential to understand the properties of their roots. The Factor Theorem is a powerful tool that can help us determine if a polynomial has a certain root. By using this theorem and evaluating the function at the suspected root, we can determine if our suspicions are correct. In this case, we found that $\frac{3}{4}$ is not a root of the function $q(x) = 6x^3 + 19x^2 - 15x - 28$.
Q&A: Understanding Polynomial Functions and Their Roots

Introduction

In our previous article, we discussed why $\frac{3}{4}$ cannot be a root of the function $q(x) = 6x^3 + 19x^2 - 15x - 28$. We used the Factor Theorem and evaluated the function at this value to determine that it is not a root. In this article, we will answer some common questions related to polynomial functions and their roots.

Q: What is a root of a polynomial function?

A: A root of a polynomial function is a value of x that makes the function equal to zero. In other words, if we have a polynomial function $f(x)$, then a root of $f(x)$ is a value of x such that $f(x) = 0$.

Q: How do we find the roots of a polynomial function?

A: There are several methods for finding the roots of a polynomial function, including factoring, the Rational Root Theorem, and the use of a graphing calculator or computer software. Factoring involves breaking down the polynomial into simpler expressions, while the Rational Root Theorem provides a list of possible rational roots.

Q: What is the Rational Root Theorem?

A: The Rational Root Theorem states that if a polynomial $f(x)$ has integer coefficients and a rational root $p/q$, where $p$ and $q$ are integers and $q$ is nonzero, then $p$ must be a factor of the constant term of $f(x)$ and $q$ must be a factor of the leading coefficient of $f(x)$.

Q: How do we use the Rational Root Theorem to find roots?

A: To use the Rational Root Theorem, we first need to identify the factors of the constant term and the leading coefficient of the polynomial. We then list all possible combinations of these factors as rational numbers. We can then test each of these rational numbers to see if it is a root of the polynomial.

Q: What is the difference between a root and a solution?

A: A root of a polynomial function is a value of x that makes the function equal to zero. A solution to a polynomial equation is a value of x that satisfies the equation. In other words, a root is a specific value of x that makes the function equal to zero, while a solution is a value of x that satisfies the equation.

Q: Can a polynomial function have multiple roots?

A: Yes, a polynomial function can have multiple roots. In fact, a polynomial function can have any number of roots, including zero, one, two, or more.

Q: How do we determine the number of roots of a polynomial function?

A: We can determine the number of roots of a polynomial function by using the Fundamental Theorem of Algebra, which states that a polynomial function of degree n has exactly n complex roots, counting multiplicities.

Q: What is the Fundamental Theorem of Algebra?

A: The Fundamental Theorem of Algebra states that a polynomial function of degree n has exactly n complex roots, counting multiplicities. In other words, if we have a polynomial function $f(x)$ of degree n, then $f(x)$ has exactly n complex roots, including real and imaginary roots.

Q: Can a polynomial function have complex roots?

A: Yes, a polynomial function can have complex roots. In fact, a polynomial function can have any number of complex roots, including real and imaginary roots.

Q: How do we find complex roots of a polynomial function?

A: We can find complex roots of a polynomial function by using the quadratic formula, the Rational Root Theorem, or other methods. We can also use a graphing calculator or computer software to find complex roots.

Q: What is the difference between a real root and a complex root?

A: A real root is a root of a polynomial function that is a real number, while a complex root is a root of a polynomial function that is a complex number. In other words, a real root is a value of x that makes the function equal to zero, while a complex root is a value of x that makes the function equal to zero, but is not a real number.

Q: Can a polynomial function have both real and complex roots?

A: Yes, a polynomial function can have both real and complex roots. In fact, a polynomial function can have any combination of real and complex roots.

Q: How do we determine if a polynomial function has real or complex roots?

A: We can determine if a polynomial function has real or complex roots by using the discriminant, the Rational Root Theorem, or other methods. We can also use a graphing calculator or computer software to determine if a polynomial function has real or complex roots.

Conclusion

In this article, we have answered some common questions related to polynomial functions and their roots. We have discussed the definition of a root, how to find roots, the Rational Root Theorem, and the difference between real and complex roots. We have also discussed how to determine if a polynomial function has real or complex roots. By understanding these concepts, we can better understand polynomial functions and their roots.