Type The Correct Answer In Each Box. Use Numerals Instead Of Words. If Necessary, Use / For The Fraction Bar(s).Find The Factors Of The Function $f(x)=2x^4-x^3-18x^2+9x$, And Use Them To Complete This Statement:From Left To Right, Function

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Introduction

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In mathematics, a polynomial function is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. Factoring a polynomial function involves expressing it as a product of simpler expressions, called factors. In this article, we will find the factors of the given polynomial function $f(x)=2x^4-x3-18x^2+9x$ and use them to complete a statement.

Understanding the Function

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The given polynomial function is $f(x)=2x^4-x3-18x^2+9x$. To find its factors, we need to identify the greatest common factor (GCF) of the terms and then use the GCF to factor out the terms.

Finding the Greatest Common Factor (GCF)

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The GCF of the terms in the polynomial function is the largest expression that divides each term without leaving a remainder. In this case, the GCF is $x$.

Factoring Out the GCF

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We can factor out the GCF, $x$, from each term in the polynomial function:

$$f(x)=2x^4-x^3-18x^2+9x = x(2x^3-x^2-18x+9)$$

Factoring the Remaining Expression

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The remaining expression, $2x^3-x^2-18x+9$, can be factored further. We can try to find two binomials whose product is equal to the remaining expression.

Using the Rational Root Theorem

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The rational root theorem states that if a rational number $p/q$ is a root of the polynomial function, then $p$ must be a factor of the constant term, and $q$ must be a factor of the leading coefficient. In this case, the constant term is 9, and the leading coefficient is 2.

Finding the Factors

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Using the rational root theorem, we can find the possible rational roots of the polynomial function. The possible rational roots are $\pm 1, \pm 3, \pm 9, \pm 1/2, \pm 3/2, \pm 9/2$.

Testing the Possible Rational Roots

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We can test each of the possible rational roots by substituting them into the polynomial function and checking if the result is equal to zero.

Finding the First Factor

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After testing the possible rational roots, we find that $x=3/2$ is a root of the polynomial function. This means that $(x-3/2)$ is a factor of the polynomial function.

Factoring the Remaining Expression

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We can factor the remaining expression by dividing it by $(x-3/2)$:

$$2x^3-x^2-18x+9 = (x-3/2)(2x^2+3x-6)$$

Factoring the Quadratic Expression

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The quadratic expression $2x^2+3x-6$ can be factored further:

$$2x^2+3x-6 = (2x-3)(x+2)$$

Finding the Factors of the Polynomial Function

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We have now factored the polynomial function into the following expression:

$$f(x)=2x^4-x^3-18x^2+9x = x(x-3/2)(2x-3)(x+2)$$

Completing the Statement

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From left to right, the factors of the polynomial function are:

1. $x$
2. $(x-3/2)$
3. $(2x-3)$
4. $(x+2)$

Therefore, the correct answer is:

1. $x$
2. $(x-3/2)$
3. $(2x-3)$
4. $(x+2)$
   

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Introduction

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In our previous article, we discussed how to find the factors of a polynomial function. In this article, we will answer some frequently asked questions about finding factors of a polynomial function.

Q: What is the greatest common factor (GCF) of a polynomial function?

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A: The greatest common factor (GCF) of a polynomial function is the largest expression that divides each term without leaving a remainder.

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

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A: To find the GCF of a polynomial function, you need to identify the common factors of the terms. You can do this by looking for the largest expression that divides each term without leaving a remainder.

Q: What is the rational root theorem?

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A: The rational root theorem states that if a rational number $p/q$ is a root of the polynomial function, then $p$ must be a factor of the constant term, and $q$ must be a factor of the leading coefficient.

Q: How do I use the rational root theorem to find the factors of a polynomial function?

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A: To use the rational root theorem, you need to find the possible rational roots of the polynomial function. You can do this by listing all the possible rational roots and testing them to see if they are actually roots of the polynomial function.

Q: What is the difference between a factor and a root of a polynomial function?

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A: A factor of a polynomial function is an expression that divides the polynomial function without leaving a remainder. A root of a polynomial function is a value of the variable that makes the polynomial function equal to zero.

Q: How do I factor a polynomial function?

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A: To factor a polynomial function, you need to find the greatest common factor (GCF) of the terms and then use the GCF to factor out the terms. You can also use the rational root theorem to find the possible rational roots of the polynomial function and test them to see if they are actually roots of the polynomial function.

Q: What are some common mistakes to avoid when factoring a polynomial function?

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A: Some common mistakes to avoid when factoring a polynomial function include:

• Not finding the greatest common factor (GCF) of the terms
• Not using the rational root theorem to find the possible rational roots of the polynomial function
• Not testing the possible rational roots to see if they are actually roots of the polynomial function
• Not factoring the polynomial function completely

Q: How do I know if I have factored a polynomial function completely?

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A: You can check if you have factored a polynomial function completely by multiplying the factors together and checking if the result is equal to the original polynomial function.

Q: What are some real-world applications of factoring polynomial functions?

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A: Some real-world applications of factoring polynomial functions include:

• Solving systems of equations
• Finding the maximum or minimum value of a function
• Modeling real-world phenomena, such as population growth or chemical reactions

Conclusion

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In this article, we have answered some frequently asked questions about finding factors of a polynomial function. We have discussed the greatest common factor (GCF), the rational root theorem, and some common mistakes to avoid when factoring a polynomial function. We have also discussed some real-world applications of factoring polynomial functions.