Leading Coefficient and Roots

Oct 13, 2025 by miodeya.com 125 views

Understanding Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication, and non-negative integer exponents. The general form of a polynomial function is:

f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0

where a_n is the leading coefficient, and a_0 is the constant term.

Leading Coefficient and Roots

In this problem, we are given that the leading coefficient of the polynomial function is 3, and the roots are -4, i, and 2, all with multiplicity 1. This means that the polynomial function can be written in factored form as:

f(x) = a(x - r_1)(x - r_2)(x - r_3)

where a is the leading coefficient, and r_1, r_2, and r_3 are the roots.

Factoring the Polynomial Function

To find the polynomial function, we need to multiply the factors together. Since the roots are -4, i, and 2, we can write the factors as:

(x + 4), (x - i), and (x - 2)

However, we need to be careful when multiplying complex factors. The correct way to multiply (x - i) is to use the fact that i^2 = -1.

(x - i) = (x - sqrt(-1)) = (x - sqrt(-1))(x + sqrt(-1)) = (x^2 + 1)

So, the correct factors are:

(x + 4), (x^2 + 1), and (x - 2)

Multiplying the Factors

Now, we can multiply the factors together to get the polynomial function:

f(x) = (x + 4)(x^2 + 1)(x - 2)

Expanding the product, we get:

f(x) = (x^3 + 4x^2 + x + 4)(x - 2)

Expanding further, we get:

f(x) = x^4 - 2x^3 + 4x^2 + x^3 - 8x^2 + 4x + 4x - 8

Combining like terms, we get:

f(x) = x^4 - x^3 - 4x^2 + 8x - 8

Comparing with the Options

Now, we can compare our result with the options given:

A. f(x) = 3(x + 4)(x - 1)(x - 2) B. f(x) = (x - 3)(x + 4)(x - 1)(x - 2) C. f(x) = (x + 4)(x^2 + 1)(x - 2)

Our result is not equal to option A, since the leading coefficient is not 3. Our result is not equal to option B, since the leading coefficient is not 1. Our result is equal to option C, since the factors are the same.

Conclusion

Therefore, the polynomial function with a leading coefficient of 3 and roots -4, i, and 2, all with multiplicity 1, is:

f(x) = 3(x + 4)(x - 1)(x - 2)

This is option A.

Discussion

This problem requires a good understanding of polynomial functions and their properties. The key concept is to recognize that the roots are given with multiplicity 1, which means that the factors are distinct. The problem also requires careful manipulation of complex factors, using the fact that i^2 = -1.

Practice Problems

Find the polynomial function with a leading coefficient of 2 and roots 1, 2, and 3, all with multiplicity 1.
Find the polynomial function with a leading coefficient of 1 and roots -1, i, and 2, all with multiplicity 1.
Find the polynomial function with a leading coefficient of 3 and roots -2, 1, and 4, all with multiplicity 1.

Q: What is a polynomial function?

A: A polynomial function is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication, and non-negative integer exponents.

Q: What is the leading coefficient of a polynomial function?

A: The leading coefficient of a polynomial function is the coefficient of the highest degree term.

Q: What are the roots of a polynomial function?

A: The roots of a polynomial function are the values of x that make the function equal to zero.

Q: What is the multiplicity of a root?

A: The multiplicity of a root is the number of times the root appears in the factorization of the polynomial function.

Q: How do I find the polynomial function with a given leading coefficient and roots?

A: To find the polynomial function with a given leading coefficient and roots, you need to multiply the factors together. The factors are of the form (x - r), where r is a root.

Q: What if the roots are complex numbers?

A: If the roots are complex numbers, you need to use the fact that i^2 = -1 to simplify the factors.

Q: How do I expand the product of the factors?

A: To expand the product of the factors, you need to multiply each term in one factor by each term in the other factors.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the terms with the same variable and exponent.

Q: What if the polynomial function has a degree greater than 3?

A: If the polynomial function has a degree greater than 3, you need to use the fact that the product of the factors is equal to the polynomial function.

Q: How do I find the polynomial function with a given leading coefficient and roots, all with multiplicity 1?

A: To find the polynomial function with a given leading coefficient and roots, all with multiplicity 1, you need to multiply the factors together, using the fact that the roots are distinct.

Q: What if the polynomial function has a leading coefficient of 0?

A: If the polynomial function has a leading coefficient of 0, it is not a polynomial function.

Q: How do I determine if a function is a polynomial function?

A: To determine if a function is a polynomial function, you need to check if it can be written in the form f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where a_n is the leading coefficient, and a_0 is the constant term.

Q: What are some common mistakes to avoid when working with polynomial functions?

A: Some common mistakes to avoid when working with polynomial functions include:

Not using the correct order of operations
Not simplifying complex factors
Not combining like terms
Not checking if the function is a polynomial function

Q: How do I apply polynomial functions in real-world problems?

A: Polynomial functions are used in a wide range of real-world problems, including:

Modeling population growth
Describing the motion of objects
Analyzing data
Solving optimization problems

Q: What are some advanced topics in polynomial functions?

A: Some advanced topics in polynomial functions include:

Polynomial division
Synthetic division
Polynomial long division
Rational root theorem
Descartes' rule of signs

Q: How do I practice working with polynomial functions?

A: To practice working with polynomial functions, you can try the following:

Solve polynomial function problems
Graph polynomial functions
Analyze data using polynomial functions
Apply polynomial functions to real-world problems

Q: What are some resources for learning more about polynomial functions?

A: Some resources for learning more about polynomial functions include:

Textbooks
Online tutorials
Video lectures
Practice problems
Real-world applications