Dividing the Function by the Known Factor

Oct 13, 2025 by windailys.com 145 views

Introduction

In algebra, a cubic function is a polynomial function of degree three, which means the highest power of the variable is three. These functions can be written in the form $f(x) = ax^3 + bx^2 + cx + d$ , where $a$ , $b$ , $c$ , and $d$ are constants. Given a cubic function, we can find its roots, which are the values of $x$ that make the function equal to zero. In this article, we will explore how to find all the roots of a cubic function when one factor is known.

The Given Function

The given function is $f(x) = 4x^3 - 4x^2 - 16x + 16$ . We are told that one factor of this function is $(x - 2)$ . This means that when we substitute $x = 2$ into the function, the result should be zero.

Substituting $x = 2$ into the Function

Let's substitute $x = 2$ into the function $f(x) = 4x^3 - 4x^2 - 16x + 16$ .

$f(2) = 4(2)^3 - 4(2)^2 - 16(2) + 16$

$f(2) = 4(8) - 4(4) - 32 + 16$

$f(2) = 32 - 16 - 32 + 16$

$f(2) = 0$

As expected, when we substitute $x = 2$ into the function, the result is zero. This confirms that $(x - 2)$ is indeed a factor of the function.

Dividing the Function by the Known Factor

Now that we know $(x - 2)$ is a factor of the function, we can divide the function by this factor to find the other factors. To do this, we will use polynomial long division.

Polynomial Long Division

To divide $f(x) = 4x^3 - 4x^2 - 16x + 16$ by $(x - 2)$ , we will use polynomial long division.

  ____________________
x - 2 | 4x^3 - 4x^2 - 16x + 16
  -4x^3 + 8x^2
  ____________________
       -12x^2 - 16x + 16
       12x^2 - 24x
  ____________________
              8x + 16
              -8x - 16
  ____________________
                     0

The result of the division is $f(x) = (x - 2)(4x^2 + 8x + 8)$ .

Finding the Other Factors

Now that we have the result of the division, we can find the other factors of the function. The other factor is $4x^2 + 8x + 8$ . To find the roots of this quadratic function, we can use the quadratic formula.

The Quadratic Formula

The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case, $a = 4$ , $b = 8$ , and $c = 8$ . Plugging these values into the formula, we get:

$x = \frac{-8 \pm \sqrt{8^2 - 4(4)(8)}}{2(4)}$

$x = \frac{-8 \pm \sqrt{64 - 128}}{8}$

$x = \frac{-8 \pm \sqrt{-64}}{8}$

$x = \frac{-8 \pm 8i}{8}$

$x = \frac{-1 \pm i}{1}$

The roots of the quadratic function are $x = -1 + i$ and $x = -1 - i$ .

Conclusion

In conclusion, we have found all the roots of the given cubic function. The roots are $x = 2$ , $x = -1 + i$ , and $x = -1 - i$ . Therefore, the correct answer is:

The Final Answer

The final answer is $\boxed{B}$ .

Explanation

$The final answer is $\boxed{B}$ because the roots of the function are $x = -2$, $x = 1$, and $x = 2$.<br/> **One Factor of a Cubic Function: Finding All Roots - Q&A**$

Introduction

In our previous article, we explored how to find all the roots of a cubic function when one factor is known. We used polynomial long division to divide the function by the known factor and then used the quadratic formula to find the roots of the resulting quadratic function. In this article, we will answer some common questions related to finding the roots of a cubic function.

Q: What is the first step in finding the roots of a cubic function?

A: The first step in finding the roots of a cubic function is to identify one factor of the function. This can be done by substituting a value of $x$ into the function and checking if the result is zero.

Q: How do I know if a value of $x$ is a root of the function?

A: If you substitute a value of $x$ into the function and the result is zero, then that value of $x$ is a root of the function.

Q: What is polynomial long division?

A: Polynomial long division is a method of dividing a polynomial by another polynomial. It is used to find the quotient and remainder of the division.

Q: How do I use polynomial long division to find the roots of a cubic function?

A: To use polynomial long division to find the roots of a cubic function, you need to divide the function by the known factor. The result of the division will be a quadratic function, which you can then use the quadratic formula to find the roots.

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to find the roots of a quadratic function. It is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Q: How do I use the quadratic formula to find the roots of a quadratic function?

A: To use the quadratic formula to find the roots of a quadratic function, you need to plug in the values of $a$ , $b$ , and $c$ into the formula. The result will be two roots, which are the values of $x$ that make the function equal to zero.

Q: What are the roots of a cubic function?

A: The roots of a cubic function are the values of $x$ that make the function equal to zero. They can be real or complex numbers.

Q: Can a cubic function have three real roots?

A: Yes, a cubic function can have three real roots. However, it is also possible for a cubic function to have one real root and two complex roots.

Q: Can a cubic function have two real roots and one complex root?

A: Yes, a cubic function can have two real roots and one complex root.

Q: Can a cubic function have one real root and two complex roots?

A: Yes, a cubic function can have one real root and two complex roots.

Conclusion

In conclusion, finding the roots of a cubic function can be a challenging task, but with the right tools and techniques, it can be done. We hope that this article has provided you with a better understanding of how to find the roots of a cubic function and has answered some of the common questions related to this topic.

Frequently Asked Questions

Q: What is the difference between a cubic function and a quadratic function? A: A cubic function is a polynomial function of degree three, while a quadratic function is a polynomial function of degree two.
Q: Can a cubic function have a negative leading coefficient? A: Yes, a cubic function can have a negative leading coefficient.
Q: Can a cubic function have a leading coefficient of zero? A: No, a cubic function cannot have a leading coefficient of zero.
Q: Can a cubic function have a constant term of zero? A: Yes, a cubic function can have a constant term of zero.

Glossary

Cubic function: A polynomial function of degree three.
Quadratic function: A polynomial function of degree two.
Polynomial long division: A method of dividing a polynomial by another polynomial.
Quadratic formula: A formula used to find the roots of a quadratic function.
Roots: The values of $x$ that make the function equal to zero.