Substituting 2 into the Function

Oct 13, 2025 by talynt.org 79 views

Introduction

In mathematics, a zero of a function is a value of the variable that makes the function equal to zero. To determine whether 2 is a zero of the function f(x) = x^3 - 11x^2 + 13x + 6, we can use the substitution method. This method involves substituting the value of x into the function and then simplifying the expression to see if it equals zero.

The Substitution Method

The substitution method is a simple and effective way to determine whether a value is a zero of a function. To use this method, we substitute the value of x into the function and then simplify the expression. If the expression equals zero, then the value is a zero of the function.

Substituting 2 into the Function

To determine whether 2 is a zero of the function f(x) = x^3 - 11x^2 + 13x + 6, we substitute 2 into the function in place of x. This gives us:

f(2) = (2)^3 - 11(2)^2 + 13(2) + 6

Simplifying the Expression

To simplify the expression, we need to follow the order of operations (PEMDAS):

Evaluate the exponents: (2)^3 = 8 and (2)^2 = 4
Multiply 11 by 4: 11(4) = 44
Multiply 13 by 2: 13(2) = 26
Substitute the values back into the expression: f(2) = 8 - 44 + 26 + 6

Evaluating the Expression

Now that we have simplified the expression, we can evaluate it to see if it equals zero:

f(2) = 8 - 44 + 26 + 6 f(2) = -4

Conclusion

Since f(2) = -4 and not 0, we can conclude that 2 is not a zero of the function f(x) = x^3 - 11x^2 + 13x + 6.

Why the Substitution Method Works

The substitution method works because it allows us to plug in a value of x into the function and see if it equals zero. If the value is a zero of the function, then the expression will equal zero. If the value is not a zero of the function, then the expression will not equal zero.

Real-World Applications

The substitution method has many real-world applications. For example, it can be used to determine whether a value is a zero of a function in physics, engineering, and economics. It can also be used to solve systems of equations and to find the roots of a polynomial.

Common Mistakes

When using the substitution method, there are several common mistakes to avoid. These include:

Not following the order of operations (PEMDAS)
Not simplifying the expression correctly
Not evaluating the expression correctly

Tips and Tricks

Here are some tips and tricks to help you use the substitution method effectively:

Make sure to follow the order of operations (PEMDAS)
Simplify the expression as much as possible
Evaluate the expression carefully
Check your work to make sure you have not made any mistakes

Conclusion

In conclusion, the substitution method is a powerful tool for determining whether a value is a zero of a function. By substituting the value into the function and simplifying the expression, we can see if it equals zero. If the value is a zero of the function, then the expression will equal zero. If the value is not a zero of the function, then the expression will not equal zero. With practice and patience, you can master the substitution method and use it to solve a wide range of mathematical problems.

Final Answer

The final answer is: A. No

Introduction

In our previous article, we used the substitution method to determine whether 2 is a zero of the function f(x) = x^3 - 11x^2 + 13x + 6. We found that 2 is not a zero of the function. In this article, we will answer some common questions related to the substitution method and provide additional examples to help you understand the concept better.

Q: What is the substitution method?

A: The substitution method is a technique used to determine whether a value is a zero of a function. It involves substituting the value into the function and simplifying the expression to see if it equals zero.

Q: How do I use the substitution method?

A: To use the substitution method, follow these steps:

Substitute the value into the function in place of x.
Simplify the expression by following the order of operations (PEMDAS).
Evaluate the expression to see if it equals zero.

Q: What if the expression does not equal zero?

A: If the expression does not equal zero, then the value is not a zero of the function.

Q: Can I use the substitution method with any type of function?

A: Yes, you can use the substitution method with any type of function, including polynomial, rational, and trigonometric functions.

Q: What are some common mistakes to avoid when using the substitution method?

A: Some common mistakes to avoid when using the substitution method include:

Not following the order of operations (PEMDAS)
Not simplifying the expression correctly
Not evaluating the expression correctly

Q: How can I practice using the substitution method?

A: You can practice using the substitution method by working through examples and exercises. Start with simple functions and gradually move on to more complex ones.

Q: What are some real-world applications of the substitution method?

A: The substitution method has many real-world applications, including:

Determining whether a value is a zero of a function in physics, engineering, and economics
Solving systems of equations
Finding the roots of a polynomial

Q: Can I use the substitution method to solve systems of equations?

A: Yes, you can use the substitution method to solve systems of equations. By substituting one equation into the other, you can solve for the variables.

Q: What are some tips and tricks for using the substitution method effectively?

A: Some tips and tricks for using the substitution method effectively include:

Make sure to follow the order of operations (PEMDAS)
Simplify the expression as much as possible
Evaluate the expression carefully
Check your work to make sure you have not made any mistakes

Q: How can I determine whether a value is a zero of a function using the substitution method?

A: To determine whether a value is a zero of a function using the substitution method, follow these steps:

Substitute the value into the function in place of x.
Simplify the expression by following the order of operations (PEMDAS).
Evaluate the expression to see if it equals zero.

Conclusion

In conclusion, the substitution method is a powerful tool for determining whether a value is a zero of a function. By substituting the value into the function and simplifying the expression, we can see if it equals zero. With practice and patience, you can master the substitution method and use it to solve a wide range of mathematical problems.

Final Answer

The final answer is: A. No

Additional Examples

Here are some additional examples to help you understand the concept better:

Determine whether 3 is a zero of the function f(x) = x^2 - 4x + 3.
Determine whether 2 is a zero of the function f(x) = x^3 - 2x^2 - 5x + 6.
Determine whether 1 is a zero of the function f(x) = x^2 - 2x - 3.

Solutions

Here are the solutions to the additional examples:

f(3) = (3)^2 - 4(3) + 3 = 9 - 12 + 3 = 0. Therefore, 3 is a zero of the function f(x) = x^2 - 4x + 3.
f(2) = (2)^3 - 2(2)^2 - 5(2) + 6 = 8 - 8 - 10 + 6 = -4. Therefore, 2 is not a zero of the function f(x) = x^3 - 2x^2 - 5x + 6.
f(1) = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4. Therefore, 1 is not a zero of the function f(x) = x^2 - 2x - 3.