Problem 7
Question
Use the Rational Zero Theorem to list all possible rational zeros for each given function. $$ f(x)=x^{5}-x^{4}-7 x^{3}+7 x^{2}-12 x-12 $$
Step-by-Step Solution
Verified Answer
The possible rational zeros of function \( f(x) = x^{5}-x^{4}-7 x^{3}+7 x^{2}-12 x-12 \) are ±1, ±2, ±3, ±4, ±6, ±12.
1Step 1: Identify the leading coefficient and constant term
In this case, coefficient of \( x^{5} \) is 1 and the constant term is -12.
2Step 2: Determine all factors
The factors of the constant term 12 are ±1, ±2, ±3, ±4, ±6, ±12. Since the leading coefficient is 1, the factors of the leading coefficient is only ±1.
3Step 3: Construct the rational zeroes
According to Rational Zero Theorem, the possible rational zeros will be the factors of constant term divided by factors of leading coefficient. Here, the possible rational zeros are all factors of -12 divided by 1, or ±1, ±2, ±3, ±4, ±6, ±12.
Key Concepts
Polynomial FunctionsRational ZerosFinding Rational ZerosLeading Coefficient and Constant Term
Polynomial Functions
Polynomial functions are mathematical expressions that consist of variables and coefficients combined using only addition, subtraction, multiplication, and non-negative integer exponents. For instance, f(x) = x^5 - x^4 - 7x^3 + 7x^2 - 12x - 12 is a polynomial function where x represents the variable, and the exponents are all non-negative integers. These functions are foundational in algebra and calculus, representing a wide range of phenomena in science and engineering.
When discussing polynomial functions, terms such as 'degree' and 'leading coefficient' are vital. The degree of a polynomial function is the highest exponent in the function. In our example, the degree is 5, because of the term x^5. The leading coefficient is the coefficient of the term with the highest exponent, which for the given function is 1.
When discussing polynomial functions, terms such as 'degree' and 'leading coefficient' are vital. The degree of a polynomial function is the highest exponent in the function. In our example, the degree is 5, because of the term x^5. The leading coefficient is the coefficient of the term with the highest exponent, which for the given function is 1.
Rational Zeros
Rational zeros, also known as rational roots, of a polynomial function, are the values of x that make the function equal to zero. These zeros are called 'rational' because they can be expressed as a fraction of two integers, where the numerator is a factor of the constant term and the denominator is a factor of the leading coefficient.
Understanding rational zeros is crucial because they provide insight into the behavior of polynomial functions, such as where the graph of the function crosses or touches the x-axis. Identifying these zeros can also help in factors or completely solving polynomial equations.
Understanding rational zeros is crucial because they provide insight into the behavior of polynomial functions, such as where the graph of the function crosses or touches the x-axis. Identifying these zeros can also help in factors or completely solving polynomial equations.
Finding Rational Zeros
The Rational Zero Theorem provides a systematic way to find all possible rational zeros of a polynomial function. This theorem states that if a polynomial has rational zeros, then each zero p/q is such that p is a factor of the constant term and q is a factor of the leading coefficient.
To find the rational zeros of the function f(x) = x^5 - x^4 - 7x^3 + 7x^2 - 12x - 12, we determine all factors of the constant term (-12) and the leading coefficient (1). We use these factors to construct a list of all possible rational zeros. It's important to consider both positive and negative factors, resulting in potential zeros such as ±1, ±2, ±3, ±4, ±6, and ±12. Testing these possible zeros in the polynomial function can confirm which are actual zeros.
To find the rational zeros of the function f(x) = x^5 - x^4 - 7x^3 + 7x^2 - 12x - 12, we determine all factors of the constant term (-12) and the leading coefficient (1). We use these factors to construct a list of all possible rational zeros. It's important to consider both positive and negative factors, resulting in potential zeros such as ±1, ±2, ±3, ±4, ±6, and ±12. Testing these possible zeros in the polynomial function can confirm which are actual zeros.
Leading Coefficient and Constant Term
In a polynomial function, the leading coefficient and the constant term play a significant role in determining its properties, including its end behavior and the Rational Zero Theorem. The constant term of a polynomial is the term with no variable part, which is -12 in the example. It is crucial because its factors are potential numerators for rational zeros.
The leading coefficient, here 1 from the term x^5, influences the shape of the polynomial's graph and its potential denominators for rational zeros according to the Rational Zero Theorem. Since every integer is a factor of 1, the possible rational zeros of this function are simply the factors of the constant term. When the leading coefficient is not 1, the number of possible rational zeros increases because one must consider various combinations of factors from both the leading coefficient and the constant term.
The leading coefficient, here 1 from the term x^5, influences the shape of the polynomial's graph and its potential denominators for rational zeros according to the Rational Zero Theorem. Since every integer is a factor of 1, the possible rational zeros of this function are simply the factors of the constant term. When the leading coefficient is not 1, the number of possible rational zeros increases because one must consider various combinations of factors from both the leading coefficient and the constant term.
Other exercises in this chapter
Problem 7
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