(a) Find The Rational Zeros And Then The Other Zeros Of The Polynomial Function $f(x) = X^3 + 3x^2 - 11x - 33$, That Is, Solve $f(x) = 0$.(b) Factor $f(x$\] Into Linear Factors.(a) Select The Correct Choice Below And Fill In

Feb 28, 2025 by ADMIN 225 views

**Rational Zeros and Factorization of Polynomial Functions**

Introduction

In this article, we will explore the process of finding the rational zeros and other zeros of a given polynomial function, as well as factoring the polynomial into linear factors. We will use the polynomial function $f(x) = x^3 + 3x^2 - 11x - 33$ as an example to demonstrate these concepts.

Rational Zeros

To find the rational zeros of a polynomial function, we can use the Rational Zero Theorem. This theorem states that if $p/q$ is a rational zero of the polynomial function $f(x)$ , where $p$ and $q$ are integers, then $p$ must be a factor of the constant term of the polynomial, and $q$ must be a factor of the leading coefficient of the polynomial.

Factors of the Constant Term

The constant term of the polynomial function $f(x) = x^3 + 3x^2 - 11x - 33$ is -33. The factors of -33 are:

±1
±3
±11
±33

Factors of the Leading Coefficient

The leading coefficient of the polynomial function $f(x) = x^3 + 3x^2 - 11x - 33$ is 1. The factors of 1 are:

Possible Rational Zeros

Using the factors of the constant term and the leading coefficient, we can find the possible rational zeros of the polynomial function. These are:

±1
±3
±11
±33

Finding Rational Zeros

To find the rational zeros of the polynomial function, we can use synthetic division or long division. Let's use synthetic division to find the rational zeros.

	1	3	-11	-33
1	1	4	-8	-40
3	1	7	-4	-12
11	1	14	3	33
33	1	36	32	1081

From the synthetic division table, we can see that the polynomial function $f(x) = x^3 + 3x^2 - 11x - 33$ has a rational zero at $x = -3$ .

Other Zeros

To find the other zeros of the polynomial function, we can use the fact that the polynomial function has a rational zero at $x = -3$ . We can factor the polynomial function as:

$f(x) = (x + 3)(x^2 - 4x + 11)$

The quadratic factor $x^2 - 4x + 11$ has no real zeros, so the other zeros of the polynomial function are complex.

Complex Zeros

To find the complex zeros of the polynomial function, we can use the quadratic formula:

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

In this case, $a = 1$ , $b = -4$ , and $c = 11$ . Plugging these values into the quadratic formula, we get:

$x = \frac{4 \pm \sqrt{(-4)^2 - 4(1)(11)}}{2(1)}$ $x = \frac{4 \pm \sqrt{16 - 44}}{2}$ $x = \frac{4 \pm \sqrt{-28}}{2}$ $x = \frac{4 \pm 2i\sqrt{7}}{2}$ $x = 2 \pm i\sqrt{7}$

Therefore, the complex zeros of the polynomial function are $x = 2 + i\sqrt{7}$ and $x = 2 - i\sqrt{7}$ .

Factorization

To factor the polynomial function $f(x) = x^3 + 3x^2 - 11x - 33$ into linear factors, we can use the fact that the polynomial function has a rational zero at $x = -3$ . We can factor the polynomial function as:

$f(x) = (x + 3)(x^2 - 4x + 11)$

The quadratic factor $x^2 - 4x + 11$ cannot be factored further into linear factors, so the polynomial function $f(x) = x^3 + 3x^2 - 11x - 33$ cannot be factored into linear factors.

Conclusion

In this article, we have explored the process of finding the rational zeros and other zeros of a given polynomial function, as well as factoring the polynomial into linear factors. We have used the polynomial function $f(x) = x^3 + 3x^2 - 11x - 33$ as an example to demonstrate these concepts. The rational zeros of the polynomial function are $x = -3$ , and the complex zeros are $x = 2 + i\sqrt{7}$ and $x = 2 - i\sqrt{7}$ . The polynomial function $f(x) = x^3 + 3x^2 - 11x - 33$ cannot be factored into linear factors.

References

Rational Zero Theorem. (n.d.). Retrieved from https://www.mathopenref.com/rationalzeros.html
Synthetic Division. (n.d.). Retrieved from https://www.mathsisfun.com/algebra/synthetic-division.html
Quadratic Formula. (n.d.). Retrieved from https://www.mathsisfun.com/algebra/quadratic-formula.html
Q&A: Rational Zeros and Factorization of Polynomial Functions ===========================================================

Q: What is the Rational Zero Theorem?

A: The Rational Zero Theorem is a theorem that states that if $p/q$ is a rational zero of the polynomial function $f(x)$ , where $p$ and $q$ are integers, then $p$ must be a factor of the constant term of the polynomial, and $q$ must be a factor of the leading coefficient of the polynomial.

Q: How do I find the rational zeros of a polynomial function?

A: To find the rational zeros of a polynomial function, you can use the Rational Zero Theorem. First, find the factors of the constant term and the leading coefficient. Then, use synthetic division or long division to test each possible rational zero.

Q: What is synthetic division?

A: Synthetic division is a method of dividing a polynomial function by a linear factor. It is a shortcut for long division and can be used to find the rational zeros of a polynomial function.

Q: How do I use synthetic division to find the rational zeros of a polynomial function?

A: To use synthetic division to find the rational zeros of a polynomial function, follow these steps:

Write down the coefficients of the polynomial function in a row.
Choose a possible rational zero and write it down on the left side of the row.
Multiply the chosen rational zero by the first coefficient and write the result below the row.
Add the result to the second coefficient and write the result below the row.
Repeat steps 3 and 4 until you reach the last coefficient.
If the result is zero, then the chosen rational zero is a rational zero of the polynomial function.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to find the zeros of a quadratic polynomial 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 zeros of a quadratic polynomial function?

A: To use the quadratic formula to find the zeros of a quadratic polynomial function, follow these steps:

Write down the coefficients of the quadratic polynomial function in the form $ax^2 + bx + c$ .
Plug the coefficients into the quadratic formula.
Simplify the expression and solve for $x$ .

Q: Can a polynomial function be factored into linear factors?

A: Not always. A polynomial function can be factored into linear factors if it has a rational zero. However, if the polynomial function has no rational zeros, it may not be possible to factor it into linear factors.

Q: What are the complex zeros of a polynomial function?

A: The complex zeros of a polynomial function are the zeros that are not real numbers. They are given by the quadratic formula and are in the form $x = a + bi$ , where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q: How do I find the complex zeros of a polynomial function?

A: To find the complex zeros of a polynomial function, use the quadratic formula and simplify the expression. The result will be a complex number in the form $x = a + bi$ .

Q: What is the difference between a rational zero and a complex zero?

A: A rational zero is a zero that is a rational number, i.e., a number that can be expressed as the ratio of two integers. A complex zero is a zero that is a complex number, i.e., a number that can be expressed in the form $a + bi$ , where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q: Can a polynomial function have both rational and complex zeros?

A: Yes, a polynomial function can have both rational and complex zeros. In fact, most polynomial functions have both rational and complex zeros.

Q: How do I determine the number of zeros of a polynomial function?

A: To determine the number of zeros of a polynomial function, use the following steps:

Find the degree of the polynomial function.
If the degree is even, the number of zeros is equal to the degree.
If the degree is odd, the number of zeros is equal to the degree minus one.

Q: What is the relationship between the zeros of a polynomial function and its graph?

A: The zeros of a polynomial function are the points on the graph where the function intersects the x-axis. The graph of a polynomial function is a curve that passes through the zeros.

Q: Can a polynomial function have multiple zeros?

A: Yes, a polynomial function can have multiple zeros. In fact, most polynomial functions have multiple zeros.

Q: How do I find the multiple zeros of a polynomial function?

A: To find the multiple zeros of a polynomial function, use the following steps:

Find the rational zeros of the polynomial function.
Use synthetic division or long division to test each rational zero.
If a rational zero is a multiple zero, it will have a multiplicity greater than one.

Q: What is the multiplicity of a zero?

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

Q: How do I determine the multiplicity of a zero?

A: To determine the multiplicity of a zero, use the following steps:

Find the rational zeros of the polynomial function.
Use synthetic division or long division to test each rational zero.
If a rational zero is a multiple zero, its multiplicity will be greater than one.

Q: Can a polynomial function have a zero with a multiplicity of zero?

A: No, a polynomial function cannot have a zero with a multiplicity of zero. The multiplicity of a zero is always greater than or equal to one.