Textbook Question
In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=x^5−x^4−7x^3+7x^2−12x−12
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4. Polynomial Functions
Zeros of Polynomial Functions
3:41 minutesProblem 20
Textbook Question
Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1. 5x^4+16x^3-15x^2+8x+16; x+4
Verified step by step guidance1
Identify the divisor from the polynomial \(x+4\). The root to test using the Factor Theorem is \(x = -4\).
Set up synthetic division using \(-4\) as the divisor and the coefficients of the polynomial \(5x^4 + 16x^3 - 15x^2 + 8x + 16\), which are \([5, 16, -15, 8, 16]\).
Perform synthetic division: bring down the first coefficient (5), multiply it by \(-4\), and add to the next coefficient. Repeat this process for each coefficient.
Check the remainder from the synthetic division. If the remainder is 0, then \(x+4\) is a factor of the polynomial.
Conclude whether \(x+4\) is a factor based on the remainder. If the remainder is 0, it confirms that \(x+4\) is a factor; otherwise, it is not.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factor Theorem
The Factor Theorem states that a polynomial f(x) has a factor (x - c) if and only if f(c) = 0. This theorem is essential for determining whether a given polynomial is a factor of another polynomial. In this context, we will evaluate the first polynomial at c = -4 to see if it equals zero, indicating that (x + 4) is a factor.
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Synthetic Division
Synthetic division is a simplified method for dividing a polynomial by a linear factor of the form (x - c). It involves using the coefficients of the polynomial and performing a series of arithmetic operations to find the quotient and remainder. This technique is particularly useful for quickly determining if a polynomial is divisible by another without performing long division.
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Polynomial Functions
Polynomial functions are mathematical expressions involving variables raised to whole number powers, combined using addition, subtraction, and multiplication. Understanding the structure of polynomial functions is crucial for applying the Factor Theorem and synthetic division. In this case, the first polynomial is a fourth-degree polynomial, which influences the behavior and roots of the function.
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