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Magazine Chapter 3: Problem 738
Use the Rational Zero Theorem to help you find the zeros of the polynomial functions. $$f(x)=2 x^{3}+5 x^{2}-6 x-9$$
Short Answer
Expert verified
The zeros of the polynomial are: \(x = -1, x = 1.5, x = -3\).
Step by step solution
01
Identify Possible Rational Zeros
According to the Rational Zero Theorem, any rational zero of the polynomial \( f(x) = 2x^3 + 5x^2 - 6x - 9 \) can be expressed as \( \frac{p}{q} \), where \( p \) is a factor of the constant term (-9) and \( q \) is a factor of the leading coefficient (2). The factors of -9 are 卤1, 卤3, 卤9, and the factors of 2 are 卤1, 卤2. Therefore, the possible rational zeros are 卤1, 卤3, 卤9, 卤\frac{1}{2}, 卤\frac{3}{2}, 卤\frac{9}{2}.
02
Test Possible Zeros Using Synthetic Division
Start testing potential rational zeros using synthetic division. Begin with zero candidates like 卤1. Use synthetic division to see if the remainder is zero, indicating the divisor is a zero of the polynomial.
03
Calculate Synthetic Division for x = -1
Perform synthetic division with \( x = -1 \). Write the coefficients: 2, 5, -6, -9. Bring down the 2, multiply by -1 and add to the next coefficient (5), resulting in 3. Multiply by -1 and add to -6, resulting in -3. Multiply by -1, add to -9, and get 0. The remainder is 0, thus \( x = -1 \) is a zero.
04
Form the Depressed Polynomial
Since \( x = -1 \) is a zero, the polynomial can be factored as \( (x + 1)(2x^2 + 3x - 9) \). Now focus on finding the zeros of the quadratic \( 2x^2 + 3x - 9 \).
05
Solve the Quadratic Equation
Solve the quadratic equation \( 2x^2 + 3x - 9 = 0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = 3 \), and \( c = -9 \). Begin by calculating the discriminant: \( 3^2 - 4 \times 2 \times (-9) = 9 + 72 = 81 \). Now, find the roots: \( x = \frac{-3 \pm \sqrt{81}}{4} = \frac{-3 \pm 9}{4} \). This gives us \( x = 1.5 \) and \( x = -3 \).
06
List All Zeros of the Polynomial
The zeros of the polynomial \( f(x) = 2x^3 + 5x^2 - 6x - 9 \) are \( x = -1, x = 1.5, x = -3 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Functions
A polynomial function is an expression composed of variables and coefficients that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
For example, the function \( f(x) = 2x^3 + 5x^2 - 6x - 9 \) discussed in our exercise is a polynomial. Let's break it down:
For example, the function \( f(x) = 2x^3 + 5x^2 - 6x - 9 \) discussed in our exercise is a polynomial. Let's break it down:
- The highest degree, which is the greatest exponent (3 in this case), gives the polynomial its designation: cubic.
- Each term, like \(2x^3\) or \(-9\), consists of a coefficient and a variable part.
- The constant term is the term without a variable, here \(-9\).
Synthetic Division
Synthetic division is a simplified form of long division used to divide polynomials, especially when verifying potential zero values. It's particularly straightforward when the divisor is of the form \(x - c\). Let's explore the process as applied in the example:
- Write the coefficients of the polynomial in a row: for \(f(x) = 2x^3 + 5x^2 - 6x - 9\), you list 2, 5, -6, -9.
- Bring down the lead coefficient (first number) to the bottom row.
- Multiply it by the candidate zero (here, \(x = -1\)) and add this result to the next coefficient.
- Repeat the multiply and add process for each subsequent coefficient.
- If the final result is zero, the candidate is a true zero of the polynomial.
Quadratic Equation
A quadratic equation is a second-degree polynomial with the general form \( ax^2 + bx + c = 0 \). In our polynomial exercise, once one zero of the cubic polynomial is found, the remaining factors form a quadratic polynomial, \( 2x^2 + 3x - 9 \).
To solve such an equation, the quadratic formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] is frequently used. Here's what each part means:
To solve such an equation, the quadratic formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] is frequently used. Here's what each part means:
- \(a, b, c\) are coefficients from the quadratic equation.
- The part under the square root, \(b^2 - 4ac\), is called the discriminant. It tells you about the nature of the roots.
- Using the values \(a=2\), \(b=3\), \(c=-9\), we calculate the discriminant as 81.
- Since the discriminant is positive, the quadratic equation has two real and distinct roots, \(x = 1.5\) and \(x = -3\).
