Problem 6
Question
In Exercises \(1-8,\) use the Rational Zero Theorem to list all possible rational zeros for each given function. $$ f(x)-3 x^{4}-11 x^{3}-3 x^{2}-6 x+8 $$
Step-by-Step Solution
Verified Answer
Possible rational zeros of the given function are \(\pm 1, \pm 2, \pm 4, \pm 8, \pm 1/3, \pm 2/3, \pm 4/3, \pm 8/3\).
1Step 1: Identify the constant term and the leading coefficient
In the given function \(f(x) = -3 x^{4}-11 x^{3}-3 x^{2}-6 x+8\), the constant term (or 'free' term) is 8 and the leading coefficient is -3.
2Step 2: Find all factors of the constant term and the leading coefficient
The factors of 8 (constant term) are \(\pm 1, \pm 2, \pm 4, \pm 8\). The factors of -3 (leading coefficient) are \(\pm 1, \pm 3\).
3Step 3: Create all possible rational zeros using the formula \(\pm p/q\)
The rational zeros will be \(\pm\) each factor of 8 (constant term) divided by each factor of -3 (leading coefficient). This results in these possible rational zeros: \(\pm 1/1, \pm 2/1, \pm 4/1, \pm 8/1, \pm 1/3, \pm 2/3, \pm 4/3, \pm 8/3\). After simplifying, these possible zeros are \(\pm 1, \pm 2, \pm 4, \pm 8, \pm 1/3, \pm 2/3, \pm 4/3, \pm 8/3\).
Key Concepts
Polynomial FunctionsPossible Rational ZerosLeading CoefficientConstant Term
Polynomial Functions
A polynomial function is a mathematical expression consisting of variables (also known as indeterminates) and coefficients that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An example of a polynomial function is \( f(x) = 3x^4 - 11x^3 - 3x^2 -6x + 8 \).
Polynomial functions are classified based on their degree, which is the highest power of the variable in the function. The function \( f(x) \) mentioned is a fourth-degree polynomial, because its highest power, associated with \( x \), is 4. General characteristics of polynomial functions include their smooth and continuous nature, meaning they do not have breaks, holes, or sharp corners in their graphs.
Polynomial functions are classified based on their degree, which is the highest power of the variable in the function. The function \( f(x) \) mentioned is a fourth-degree polynomial, because its highest power, associated with \( x \), is 4. General characteristics of polynomial functions include their smooth and continuous nature, meaning they do not have breaks, holes, or sharp corners in their graphs.
Possible Rational Zeros
The Rational Zero Theorem is a handy tool when working with polynomial functions, as it helps in finding the potential rational zeros or roots of a polynomial. According to this theorem, if a polynomial function with integer coefficients has any rational zeros, they can be found with the expression \( \pm p/q \) where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient.
When you are provided with a polynomial equation, like \( f(x) = 3x^4 - 11x^3 - 3x^2 -6x + 8 \) as in the original exercise, the Rational Zero Theorem is particularly useful. It tells you that rational zeros can only be fractions formed by dividing the factors of the constant term by the factors of the leading coefficient, or their negatives. This greatly reduces the numbers you need to test when seeking possible solutions to the polynomial equation.
When you are provided with a polynomial equation, like \( f(x) = 3x^4 - 11x^3 - 3x^2 -6x + 8 \) as in the original exercise, the Rational Zero Theorem is particularly useful. It tells you that rational zeros can only be fractions formed by dividing the factors of the constant term by the factors of the leading coefficient, or their negatives. This greatly reduces the numbers you need to test when seeking possible solutions to the polynomial equation.
Leading Coefficient
The leading coefficient in a polynomial function is the coefficient of the term with the highest degree. In the polynomial \( f(x) = 3x^4 - 11x^3 - 3x^2 -6x + 8 \), the leading coefficient is 3 because it is the coefficient of \( x^4 \) - the term with the highest exponent in the function.
The leading coefficient plays a vital role in determining the end behavior of the polynomial's graph: whether the function's arms rise or fall as \( x \) approaches positive or negative infinity. It also influences the number of possible positive or negative rational zeros for the function as per the Descartes' Rule of Signs, and is a key element in applying the Rational Zero Theorem.
The leading coefficient plays a vital role in determining the end behavior of the polynomial's graph: whether the function's arms rise or fall as \( x \) approaches positive or negative infinity. It also influences the number of possible positive or negative rational zeros for the function as per the Descartes' Rule of Signs, and is a key element in applying the Rational Zero Theorem.
Constant Term
The constant term of a polynomial function is the term that does not have a variable associated with it. In layman's terms, it's the standalone number in the expression. For the given polynomial \( f(x) = 3x^4 - 11x^3 - 3x^2 -6x + 8 \), the constant term is 8.
This constant term is another critical component used in conjunction with the leading coefficient to find all the possible rational zeros of the polynomial function by the Rational Zero Theorem. It's important to note that the factors of the constant term are potential numerators \( p \) in the \( \pm p/q \) formulation that generates the list of possible zeros for testing.
This constant term is another critical component used in conjunction with the leading coefficient to find all the possible rational zeros of the polynomial function by the Rational Zero Theorem. It's important to note that the factors of the constant term are potential numerators \( p \) in the \( \pm p/q \) formulation that generates the list of possible zeros for testing.
Other exercises in this chapter
Problem 6
Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x^{2
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Find the domain of each rational function. $$h(x)=\frac{x+8}{x^{2}-64}$$
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Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$ \left(6 x^{3}+17 x^{2}+27 x+20\right) \div(3 x+4) $$
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Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x^{2
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