Given The Function F ( R ) = ( R − 7 ) ( R + 5 ) ( R − 6 F(r) = (r-7)(r+5)(r-6 F ( R ) = ( R − 7 ) ( R + 5 ) ( R − 6 ]:- The F F F -intercept Is □ \square □ - The R R R -intercepts Are □ \square □

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Introduction

In mathematics, functions are used to describe the relationship between variables. A function can be represented in various forms, including polynomial, rational, and trigonometric functions. In this article, we will focus on a polynomial function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) and explore its properties, including the ff-intercept and the rr-intercepts.

The ff-Intercept

The ff-intercept is the point on the graph of the function where the value of f(r)f(r) is equal to zero. To find the ff-intercept, we need to set the function equal to zero and solve for rr. In this case, we have:

f(r)=(r−7)(r+5)(r−6)=0f(r) = (r-7)(r+5)(r-6) = 0

To find the ff-intercept, we need to find the values of rr that make the function equal to zero. We can do this by setting each factor equal to zero and solving for rr.

Solving for rr

Let's start by setting the first factor equal to zero:

(r−7)=0(r-7) = 0

Solving for rr, we get:

r=7r = 7

Now, let's set the second factor equal to zero:

(r+5)=0(r+5) = 0

Solving for rr, we get:

r=−5r = -5

Finally, let's set the third factor equal to zero:

(r−6)=0(r-6) = 0

Solving for rr, we get:

r=6r = 6

The ff-Intercept

Now that we have found the values of rr that make the function equal to zero, we can determine the ff-intercept. Since the function is equal to zero at r=7r = 7, r=−5r = -5, and r=6r = 6, the ff-intercept is 0\boxed{0}.

The rr-Intercepts

The rr-intercepts are the points on the graph of the function where the value of rr is equal to zero. To find the rr-intercepts, we need to set the function equal to zero and solve for rr. In this case, we have:

f(r)=(r−7)(r+5)(r−6)=0f(r) = (r-7)(r+5)(r-6) = 0

We have already found the values of rr that make the function equal to zero, which are r=7r = 7, r=−5r = -5, and r=6r = 6. Therefore, the rr-intercepts are 7,−5,6\boxed{7, -5, 6}.

Conclusion

In this article, we have explored the properties of the polynomial function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6). We have found the ff-intercept and the rr-intercepts, which are essential components of the function's graph. Understanding these properties is crucial in mathematics, as it allows us to analyze and interpret the behavior of functions.

Applications of the Function

The function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) has various applications in mathematics and other fields. For example, it can be used to model real-world phenomena, such as population growth or chemical reactions. Additionally, it can be used to solve systems of equations and to find the roots of polynomials.

Graphing the Function

To graph the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6), we can use the values of rr that we have found. We can plot the points (7,0)(7, 0), (−5,0)(-5, 0), and (6,0)(6, 0) on the graph, and then use the fact that the function is a polynomial to determine the shape of the graph.

Properties of the Function

The function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) has several properties that are worth noting. For example, it is a polynomial function of degree 3, which means that it has at most three roots. Additionally, it is an odd function, which means that f(−r)=−f(r)f(-r) = -f(r) for all values of rr.

Simplifying the Function

We can simplify the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) by multiplying the factors together. This gives us:

f(r)=(r−7)(r+5)(r−6)=(r2−2r−35)(r−6)f(r) = (r-7)(r+5)(r-6) = (r^2 - 2r - 35)(r-6)

We can further simplify this expression by multiplying the factors together:

f(r)=(r2−2r−35)(r−6)=r3−8r2−17r+210f(r) = (r^2 - 2r - 35)(r-6) = r^3 - 8r^2 - 17r + 210

Conclusion

In this article, we have explored the properties of the polynomial function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6). We have found the ff-intercept and the rr-intercepts, which are essential components of the function's graph. We have also simplified the function and explored its properties, including its degree and its behavior under certain transformations. Understanding these properties is crucial in mathematics, as it allows us to analyze and interpret the behavior of functions.

Final Thoughts

The function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) is a simple yet powerful tool for modeling real-world phenomena. Its properties, including its ff-intercept and rr-intercepts, are essential components of its graph. By understanding these properties, we can gain a deeper understanding of the behavior of functions and how they can be used to model real-world phenomena.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Linear Algebra" by Jim Hefferon

Glossary

  • Function: A relation between a set of inputs and a set of possible outputs.
  • Polynomial function: A function that can be written in the form f(x)=anxn+an−1xn−1+⋯+a1x+a0f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0, where an≠0a_n \neq 0.
  • Root: A value of xx that makes the function equal to zero.
  • Intercept: A point on the graph of the function where the value of the function is equal to zero.

Q: What is the ff-intercept of the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6)?

A: The ff-intercept of the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) is 0\boxed{0}.

Q: What are the rr-intercepts of the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6)?

A: The rr-intercepts of the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) are 7,−5,6\boxed{7, -5, 6}.

Q: How do I graph the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6)?

A: To graph the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6), you can plot the points (7,0)(7, 0), (−5,0)(-5, 0), and (6,0)(6, 0) on the graph, and then use the fact that the function is a polynomial to determine the shape of the graph.

Q: What are some applications of the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6)?

A: The function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) has various applications in mathematics and other fields, such as modeling real-world phenomena, solving systems of equations, and finding the roots of polynomials.

Q: Can I simplify the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6)?

A: Yes, you can simplify the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) by multiplying the factors together. This gives you:

f(r)=(r−7)(r+5)(r−6)=(r2−2r−35)(r−6)f(r) = (r-7)(r+5)(r-6) = (r^2 - 2r - 35)(r-6)

You can further simplify this expression by multiplying the factors together:

f(r)=(r2−2r−35)(r−6)=r3−8r2−17r+210f(r) = (r^2 - 2r - 35)(r-6) = r^3 - 8r^2 - 17r + 210

Q: What are some properties of the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6)?

A: The function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) is a polynomial function of degree 3, which means that it has at most three roots. Additionally, it is an odd function, which means that f(−r)=−f(r)f(-r) = -f(r) for all values of rr.

Q: How do I find the roots of the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6)?

A: To find the roots of the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6), you can set the function equal to zero and solve for rr. This gives you:

(r−7)(r+5)(r−6)=0(r-7)(r+5)(r-6) = 0

You can then solve for rr by setting each factor equal to zero and solving for rr.

Q: What is the degree of the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6)?

A: The degree of the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) is 3, which means that it has at most three roots.

Q: Is the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) an odd function?

A: Yes, the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) is an odd function, which means that f(−r)=−f(r)f(-r) = -f(r) for all values of rr.

Q: Can I use the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) to model real-world phenomena?

A: Yes, the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) can be used to model real-world phenomena, such as population growth or chemical reactions.

Q: How do I use the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) to solve systems of equations?

A: To use the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) to solve systems of equations, you can substitute the function into the system of equations and solve for the variables.

Q: Can I use the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) to find the roots of polynomials?

A: Yes, the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) can be used to find the roots of polynomials.

Q: What are some other applications of the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6)?

A: The function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) has various other applications, such as in physics, engineering, and computer science.

Q: Can I use the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) to model population growth?

A: Yes, the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) can be used to model population growth.

Q: How do I use the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) to model chemical reactions?

A: To use the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) to model chemical reactions, you can substitute the function into the chemical reaction and solve for the variables.

Q: Can I use the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) to find the roots of quadratic equations?

A: Yes, the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) can be used to find the roots of quadratic equations.

Q: What are some other properties of the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6)?

A: The function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) is a polynomial function of degree 3, which means that it has at most three roots. Additionally, it is an odd function, which means that f(−r)=−f(r)f(-r) = -f(r) for all values of rr.

Q: Can I use the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) to model real-world phenomena in physics?

A: Yes, the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) can be used to model real-world phenomena in physics.

Q: How do I use the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) to model real-world phenomena in engineering?

A: To use the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) to model real-world phenomena in engineering, you can substitute the function into the engineering problem and solve for the variables.

Q: Can I use the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) to find the roots of cubic equations?

A: Yes, the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) can be used to find the roots of cubic equations.

Q: What are some other applications of the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) in computer science?

A: The function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) has various other applications in computer science, such as in algorithms and data structures.

Q: Can I use the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) to model real-world phenomena in computer science?

A: Yes, the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) can be used to model real-world phenomena in computer science.

Q: How do I use the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) to model real-world phenomena in data analysis?

A: To use the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) to model real-world phenomena in data analysis, you can substitute the function into the data analysis problem and solve for the variables.

Q: Can I use the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) to find the roots of quartic equations?

A: Yes, the function f(r)=(r−7)(r+5)(r−6)f(r) = (r-7)(r+5)(r-6) can be used to find the roots of quartic equations.

**Q: What are