Ch. 3 - Polynomial and Rational Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 19

Chapter 4, Problem 19

In Exercises 17–24, a) List all possible rational roots. b) List all possible rational roots. c) Use the quotient from part (b) to find the remaining roots and solve the equation. x³−10x−12=0

Verified step by step guidance

Identify the polynomial given: $x^{3} - 10x - 12 = 0$. Notice that the polynomial is cubic, and the constant term is $-12$, while the leading coefficient is $1$.

Use the Rational Root Theorem to list all possible rational roots. According to the theorem, possible rational roots are of the form $\pm \frac{p}{q}$, where $p$ divides the constant term and $q$ divides the leading coefficient. Since the leading coefficient is 1, possible roots are the divisors of $-12$: $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$.

Test each possible rational root by substituting into the polynomial to find which one(s) yield zero. This can be done by direct substitution or synthetic division.

Once a root is found, use synthetic division (or polynomial long division) to divide the original polynomial by $(x - r)$, where $r$ is the root found. This will give a quadratic quotient.

Solve the quadratic quotient using factoring, completing the square, or the quadratic formula to find the remaining roots of the equation.

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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 roots of a polynomial equation 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). This theorem narrows down candidates for testing actual roots.

Polynomial Division (Synthetic or Long Division)

Polynomial division is used to divide a polynomial by a binomial of the form (x - r), where r is a root. This process simplifies the polynomial to a lower degree, making it easier to find remaining roots. Synthetic division is a streamlined method often used for this purpose.

Solving Polynomial Equations

After factoring the polynomial using known roots, the remaining polynomial can be solved by factoring further or applying methods like the quadratic formula. This step finds all roots, including irrational or complex ones, completing the solution to the equation.

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Related Practice

Textbook Question

In Exercises 19–24, (a) Use the Leading Coefficient Test to determine the graph's end behavior. (b) Determine whether the graph has y-axis symmetry, origin symmetry, or neither. (c) Graph the function. $f (x) = x^{3} - x^{2} - 9 x + 9$

Textbook Question

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $f (x) = 11 x^{3} - 6 x^{2} + x + 3$

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Textbook Question

Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20.

As $x \to \infty, f (x) \to$ _____

Textbook Question

Divide using synthetic division. (3x²+7x−20)÷(x+5)

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Textbook Question

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. $x squared plus 2 x is less than 0$

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Textbook Question

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In Exercises 17–24, a) List all possible rational roots. b) List all possible rational roots. c) Use the quotient from part (b) to find the remaining roots and solve the equation. x3−10x−12=0

Verified video answer for a similar problem:

Key Concepts

Rational Root Theorem

Polynomial Division (Synthetic or Long Division)

Solving Polynomial Equations

In Exercises 17–24, a) List all possible rational roots. b) List all possible rational roots. c) Use the quotient from part (b) to find the remaining roots and solve the equation. x³−10x−12=0