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Chapter 3: Problem 37

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=24 x^{3}+10 x^{2}-13 x-6$$

Short Answer

Expert verified
The rational zero is \( x = \frac{1}{2} \); factored form is \( (x - \frac{1}{2})(x + 1)(3x + 3)\)."

Step by step solution

01

Identify Rational Zero Theorem

The Rational Zero Theorem states that any rational solution \( \frac{p}{q} \) of the polynomial equation \( a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0 \) must be such that \( p \) is a factor of the constant term \( a_0 \), and \( q \) is a factor of the leading coefficient \( a_n \). In this polynomial, the constant term is \(-6\) and the leading coefficient is \(24\).
02

List Possible Rational Zeros

First, list the factors of the constant term \(-6\): \( \pm 1, \pm 2, \pm 3, \pm 6\). Then list the factors of the leading coefficient \(24\): \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24\). The possible rational zeros are all the fractions of these factors: \( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \ldots, \pm \frac{6}{24}\).
03

Test Possible Zeros Using Synthetic Division

Use synthetic division to test each possible rational zero. Start with realistic values like \(x = 1\), \(x = -1\), etc. Perform synthetic division to see if the remainder is zero, indicating \(x\) is a root. Through testing, we find that \(x = \frac{1}{2}\) is a root because the remainder is zero.
04

Factor Out the Zero Found

Since \(x = \frac{1}{2}\) is a zero, we can factor \(x - \frac{1}{2}\) out of the polynomial. Using synthetic division, divide \(P(x)\) by \(2x - 1\) (equivalent to multiplying through by 2 to avoid fractions), resulting in \(12x^2 + 16x + 12\).
05

Factor the Quotient Quadratic Polynomial

Now factor the quadratic polynomial \(12x^2 + 16x + 12\). Look for two numbers that multiply to \(12 \times 12 = 144\) and add up to \(16\). After testing pairs, notice that this is reducible by first reducing: \(12(x^2 + \frac{4}{3}x + 1)\), which can be further factored by completing the square or using quadratic formula. This gives \( (2x + 2)(6x + 6) \) after reduction.
06

Write Polynomial in Completely Factored Form

Finally, the fully factored form of \(P(x) = 24x^3 + 10x^2 - 13x - 6\) is \((x - \frac{1}{2})(2x + 2)(6x + 6)\). Simplifying further would give: \(2(x - \frac{1}{2})(x + 1)(3x + 3)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Synthetic Division
Synthetic division is a simplified form of polynomial division, especially useful when dividing by a linear factor. It's quicker and more systematic than traditional polynomial division. We use synthetic division primarily to find roots or zeros of a given polynomial. In synthetic division, the coefficients of the polynomial are written in a horizontal line. We then perform a series of operations to simplify the polynomial and figure out if a certain value is a root.
  • Start with the value you're testing as a possible zero. This is placed to the left of the line.
  • Write down the coefficients of the polynomial in descending order.
  • The first coefficient is brought down unchanged. Use this to multiply the leading tested zero, and write the product under the next coefficient.
  • Add down the second column, write the result, and repeat the process.
You know that the tested value is a root if the remainder, which is the last number you write in normal synthetic division, equals zero.
Factoring Polynomials
Factoring polynomials involves breaking down a polynomial into its component parts, called factors, that when multiplied give the original polynomial. It's a crucial step in simplifying expressions and solving equations. For example, with the polynomial provided, after using synthetic division, we ended up with a simpler quadratic polynomial to factor.
  • First, identify any common factors in all terms of the polynomial and factor these out.
  • If dealing with a quadratic, look for two numbers that multiply to the product of the leading coefficient and the constant term and add up to the middle term's coefficient.
  • Use various factoring techniques such as grouping, using the quadratic formula, or completing the square if necessary.
Factoring helps in simplifying problems and finding the polynomial's roots directly from its factors.
Quadratic Polynomial
A quadratic polynomial is a second-degree polynomial of the form \( ax^2 + bx + c \). It contains three terms, with the highest degree being two. In the synthetic division process, after finding a root of the original polynomial, one often ends up with a reduced quadratic equation like in the given polynomial problem.
  • Quadratic polynomials can often be factored into two binomials, possibly involving irrational numbers, or into linear factors over the complex numbers.
  • When it's difficult to factor by inspection, the quadratic formula \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \) may help find the roots.
Understanding quadratics is key, as they frequently appear both in isolation and as parts of higher-degree polynomials in mathematics.
Polynomial Roots
Polynomial roots, or zeros, are the values of \( x \) for which the polynomial equals zero. They are the solutions to the equation \( P(x) = 0 \). Finding roots is one of the essential aspects of understanding polynomial behavior because roots tell us where the graph of the polynomial crosses the x-axis.
  • The Rational Zero Theorem aids in determining possible rational roots by relating them to factors of the constant term and the leading coefficient.
  • Testing these possible values through synthetic division helps verify actual roots.
  • Once a root (or multiple roots) is found, you can factor the polynomial, which simplifies finding other roots.
Knowing how to find and interpret roots is fundamental for graphing polynomials and solving polynomial equations.

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