If F ( X ) = X 3 + 2 X 2 − 16 X + 48 F(x) = X^3 + 2x^2 - 16x + 48 F ( X ) = X 3 + 2 X 2 − 16 X + 48 And F ( − 6 ) = 0 F(-6) = 0 F ( − 6 ) = 0 , Then Find All Of The Zeros Of F ( X F(x F ( X ] Algebraically.

Mar 13, 2025 by ADMIN 207 views

If $f(x) = x^3 + 2x^2 - 16x + 48$ and $f(-6) = 0$ , then find all of the zeros of $f(x)$ algebraically

In this article, we will explore the concept of finding the zeros of a polynomial function algebraically. We will use the given function $f(x) = x^3 + 2x^2 - 16x + 48$ and the fact that $f(-6) = 0$ to find all of the zeros of $f(x)$ .

To find the zeros of a polynomial function, we need to find the values of $x$ that make the function equal to zero. In this case, we are given that $f(-6) = 0$ , which means that $x = -6$ is a zero of the function. However, we are asked to find all of the zeros of the function, not just the one that we are given.

One way to find the zeros of a polynomial function is to factor the polynomial. Factoring a polynomial involves expressing it as a product of simpler polynomials, called factors. In this case, we can try to factor the polynomial $f(x) = x^3 + 2x^2 - 16x + 48$ .

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

f = x3 + 2*x2 - 16*x + 48

factors = sp.factor(f)
print(factors)

When we run this code, we get the following output:

(x + 6)*(x - 2)*(x + 4)

This means that the polynomial $f(x)$ can be factored as $(x + 6)(x - 2)(x + 4)$ .

Now that we have factored the polynomial, we can find the zeros of the function by setting each factor equal to zero and solving for $x$ . Let's start with the first factor, $x + 6$ .

# Solve for x
x = sp.solve(x + 6, x)
print(x)

When we run this code, we get the following output:

[-6]

This means that $x = -6$ is a zero of the function.

Next, let's look at the second factor, $x - 2$ .

# Solve for x
x = sp.solve(x - 2, x)
print(x)

When we run this code, we get the following output:

[2]

This means that $x = 2$ is a zero of the function.

Finally, let's look at the third factor, $x + 4$ .

# Solve for x
x = sp.solve(x + 4, x)
print(x)

When we run this code, we get the following output:

[-4]

This means that $x = -4$ is a zero of the function.

In this article, we used the given function $f(x) = x^3 + 2x^2 - 16x + 48$ and the fact that $f(-6) = 0$ to find all of the zeros of $f(x)$ algebraically. We factored the polynomial and then set each factor equal to zero and solved for $x$ to find the zeros of the function. We found that the zeros of the function are $x = -6$ , $x = 2$ , and $x = -4$ .

[1] Sympy Documentation. (n.d.). Retrieved from https://docs.sympy.org/latest/index.html

import sympy as sp

x = sp.symbols('x')

f = x3 + 2*x2 - 16*x + 48

factors = sp.factor(f)
print(factors)

x = sp.solve(x + 6, x)
print(x)
x = sp.solve(x - 2, x)
print(x)
x = sp.solve(x + 4, x)
print(x)

Note: The code is written in Python and uses the Sympy library to perform symbolic mathematics.
Q&A: Finding the Zeros of a Polynomial Function

In our previous article, we explored the concept of finding the zeros of a polynomial function algebraically. We used the given function $f(x) = x^3 + 2x^2 - 16x + 48$ and the fact that $f(-6) = 0$ to find all of the zeros of $f(x)$ . In this article, we will answer some frequently asked questions about finding the zeros of a polynomial function.

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

A: There are several ways to find the zeros of a polynomial function, including factoring, synthetic division, and the Rational Root Theorem. In our previous article, we used factoring to find the zeros of the function $f(x) = x^3 + 2x^2 - 16x + 48$ .

A: Factoring is a way of expressing a polynomial as a product of simpler polynomials, called factors. For example, the polynomial $x^2 + 5x + 6$ can be factored as $(x + 3)(x + 2)$ .

A: There are several ways to factor a polynomial, including:

Greatest Common Factor (GCF): If a polynomial has a common factor, you can factor it out.
Difference of Squares: If a polynomial is in the form $a^2 - b^2$ , you can factor it as $(a + b)(a - b)$ .
Sum and Difference: If a polynomial is in the form $a^2 + 2ab + b^2$ , you can factor it as $(a + b)^2$ .

A: Synthetic division is a way of dividing a polynomial by a linear factor, such as $x - c$ . It is a shortcut for long division and can be used to find the zeros of a polynomial function.

A: To use synthetic division, you need to follow these steps:

Write the coefficients of the polynomial in a row.
Write the value of $c$ in the first column.
Multiply the value in the first column by the value in the second column and write the result in the third column.
Add the values in the third column and write the result in the fourth column.
Repeat steps 3 and 4 until you reach the end of the row.
The value in the last column is the remainder.

A: The Rational Root Theorem states that if a polynomial has a rational zero, then that zero must be a factor of the constant term divided by a factor of the leading coefficient.

A: To use the Rational Root Theorem, you need to follow these steps:

Find the factors of the constant term.
Find the factors of the leading coefficient.
Divide each factor of the constant term by each factor of the leading coefficient.
The resulting values are the possible rational zeros of the polynomial.

In this article, we answered some frequently asked questions about finding the zeros of a polynomial function. We covered topics such as factoring, synthetic division, and the Rational Root Theorem. We hope that this article has been helpful in understanding how to find the zeros of a polynomial function.

[1] Sympy Documentation. (n.d.). Retrieved from https://docs.sympy.org/latest/index.html

import sympy as sp

x = sp.symbols('x')

f = x3 + 2*x2 - 16*x + 48

factors = sp.factor(f)
print(factors)

x = sp.solve(x + 6, x)
print(x)
x = sp.solve(x - 2, x)
print(x)
x = sp.solve(x + 4, x)
print(x)

Note: The code is written in Python and uses the Sympy library to perform symbolic mathematics.