In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=x³−2x²−11x+12

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Identify the polynomial function: $f(x) = x^{3} - 2x^{2} - 11x + 12$.

List all possible rational zeros using the Rational Root Theorem. These are all factors of the constant term (12) divided by factors of the leading coefficient (1). So, possible rational zeros are $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$.

Use synthetic division to test each possible rational zero by substituting them into the polynomial. Start with one candidate, perform synthetic division, and check if the remainder is zero. If the remainder is zero, that candidate is an actual zero of the polynomial.

Once an actual zero is found, the quotient from the synthetic division will be a quadratic polynomial. Write down this quotient polynomial.

Solve the quadratic polynomial from the quotient using factoring, completing the square, or the quadratic formula to find the remaining zeros of the original polynomial.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rational Root Theorem

The Rational Root Theorem helps identify all possible rational zeros of a polynomial by considering factors of the constant term and the leading coefficient. These possible roots are expressed as ±(factors of constant term)/(factors of leading coefficient), providing a finite list to test.

Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a binomial of the form (x - c). It simplifies the process of evaluating whether a candidate root is an actual zero by checking if the remainder is zero, and it produces a quotient polynomial for further analysis.

Factoring Polynomials and Finding Zeros

Once a root is found using synthetic division, the quotient polynomial can be factored further or solved using other methods to find remaining zeros. This step breaks down the polynomial into simpler factors, allowing all roots (real or complex) to be identified.

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Textbook Question

In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=2x³−3x²−11x+6

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Textbook Question

Determine which functions are polynomial functions. For those that are, identify the degree. $f (x) = (x^{2} + 7) / 3$

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Textbook Question

Use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d).]

$f (x) = x^{6} - 6 x^{4} + 9 x^{2}$