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

Introduction

In mathematics, a root of a polynomial function is a value of the variable that makes the function equal to zero. When evaluating whether a given value is a root of a function, we can use various methods such as direct substitution, synthetic division, or the Rational Root Theorem. In this article, we will explore why $\frac{3}{4}$ cannot be a root of the given function $q(x) = 6x^3 + 19x^2 - 15x - 28$.

Understanding the Rational Root Theorem

The Rational Root Theorem states that if a rational number $p/q$ is a root of the polynomial function $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where $p$ and $q$ are integers and $q$ is non-zero, then $p$ must be a factor of the constant term $a_0$, and $q$ must be a factor of the leading coefficient $a_n$. In the case of the given function $q(x) = 6x^3 + 19x^2 - 15x - 28$, the leading coefficient is 6, and the constant term is -28.

Applying the Rational Root Theorem

To determine whether $\frac{3}{4}$ can be a root of the given function, we need to check if the numerator 3 is a factor of the constant term -28 and if the denominator 4 is a factor of the leading coefficient 6. However, 3 is not a factor of -28, and 4 is not a factor of 6. Therefore, according to the Rational Root Theorem, $\frac{3}{4}$ cannot be a root of the given function.

Direct Substitution

Another way to evaluate whether $\frac{3}{4}$ is a root of the given function is to substitute the value into the function and check if the result is equal to zero. We can calculate $q(\frac{3}{4})$ as follows:

$$q\left(\frac{3}{4}\right) = 6\left(\frac{3}{4}\right)^3 + 19\left(\frac{3}{4}\right)^2 - 15\left(\frac{3}{4}\right) - 28$$

Simplifying the expression, we get:

$$q\left(\frac{3}{4}\right) = 6\left(\frac{27}{64}\right) + 19\left(\frac{9}{16}\right) - 15\left(\frac{3}{4}\right) - 28$$

$$q\left(\frac{3}{4}\right) = \frac{243}{64} + \frac{171}{16} - \frac{45}{4} - 28$$

$$q\left(\frac{3}{4}\right) = \frac{243}{64} + \frac{1089}{64} - \frac{180}{16} - \frac{1792}{64}$$

$$q\left(\frac{3}{4}\right) = \frac{243 + 1089 - 720 - 1792}{64}$$

$$q\left(\frac{3}{4}\right) = \frac{720}{64}$$

$$q\left(\frac{3}{4}\right) = \frac{45}{4}$$

Since $q(\frac{3}{4})$ is not equal to zero, we can conclude that $\frac{3}{4}$ is not a root of the given function.

Conclusion

In conclusion, we have shown that $\frac{3}{4}$ cannot be a root of the given function $q(x) = 6x^3 + 19x^2 - 15x - 28$ using the Rational Root Theorem and direct substitution. The Rational Root Theorem states that if a rational number $p/q$ is a root of the polynomial function, then $p$ must be a factor of the constant term, and $q$ must be a factor of the leading coefficient. In this case, 3 is not a factor of -28, and 4 is not a factor of 6. Therefore, $\frac{3}{4}$ cannot be a root of the given function. Additionally, direct substitution confirms that $q(\frac{3}{4})$ is not equal to zero, further supporting the conclusion that $\frac{3}{4}$ is not a root of the given function.

References

• [1] "Rational Root Theorem." Wikipedia, Wikimedia Foundation, 22 Feb. 2023, en.wikipedia.org/wiki/Rational_root_theorem.
• [2] "Direct Substitution Method." Math Open Reference, mathopenref.com/directsub.html.

Further Reading

Introduction

In our previous article, we explored why $\frac{3}{4}$ cannot be a root of the given function $q(x) = 6x^3 + 19x^2 - 15x - 28$. In this article, we will answer some frequently asked questions about roots of polynomial functions, providing a deeper understanding of this mathematical concept.

Q: What is a root of a polynomial function?

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

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

A: There are several methods for finding the roots of a polynomial function, including:

• Direct substitution: Substitute values of $x$ into the function and check if the result is equal to zero.
• Synthetic division: Use synthetic division to divide the polynomial function by a linear factor.
• The Rational Root Theorem: Use the Rational Root Theorem to determine possible rational roots of the polynomial function.
• The Factor Theorem: Use the Factor Theorem to determine if a polynomial function has a specific root.

Q: What is the Rational Root Theorem?

A: The Rational Root Theorem states that if a rational number $p/q$ is a root of the polynomial function $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where $p$ and $q$ are integers and $q$ is non-zero, then $p$ must be a factor of the constant term $a_0$, and $q$ must be a factor of the leading coefficient $a_n$.

Q: How do I apply the Rational Root Theorem?

A: To apply the Rational Root Theorem, follow these steps:

1. Identify the constant term and leading coefficient of the polynomial function.
2. List all the factors of the constant term and leading coefficient.
3. Form all possible rational numbers by dividing each factor of the constant term by each factor of the leading coefficient.
4. Check if any of these rational numbers are roots of the polynomial function.

Q: What is the Factor Theorem?

A: The Factor Theorem states that if a polynomial function $f(x)$ has a root $r$, then $(x - r)$ is a factor of $f(x)$.

Q: How do I apply the Factor Theorem?

A: To apply the Factor Theorem, follow these steps:

1. Identify a root of the polynomial function.
2. Write the polynomial function as a product of the root and a quotient polynomial.
3. Factor the quotient polynomial to find the remaining factors.

Q: What are some common mistakes to avoid when finding roots of polynomial functions?

A: Some common mistakes to avoid when finding roots of polynomial functions include:

• Not checking if the rational number is a factor of the constant term and leading coefficient.
• Not using the correct method for finding roots (e.g., direct substitution, synthetic division, or the Rational Root Theorem).
• Not factoring the polynomial function correctly.
• Not checking if the quotient polynomial has any remaining factors.

Conclusion

In conclusion, finding roots of polynomial functions is an essential skill in mathematics. By understanding the Rational Root Theorem, the Factor Theorem, and other methods for finding roots, you can solve a wide range of mathematical problems. Remember to avoid common mistakes and always check your work carefully.

References

• [1] "Rational Root Theorem." Wikipedia, Wikimedia Foundation, 22 Feb. 2023, en.wikipedia.org/wiki/Rational_root_theorem.
• [2] "Factor Theorem." Math Open Reference, mathopenref.com/factor.html.
• [3] "Synthetic Division." Math Open Reference, mathopenref.com/syntheticdivision.html.

Further Reading

For more information on roots of polynomial functions, please refer to the references provided above. Additionally, you can explore other topics in mathematics, such as algebra, geometry, and calculus.