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Magazine Chapter 5: Problem 39
For the following exercises, use the Rational Zero Theorem to find all real zeros. \(8 x^{4}+26 x^{3}+39 x^{2}+26 x+6\)
Short Answer
Expert verified
Rational zeros are \(x = -\frac{1}{2}\) and \(x = -2\).
Step by step solution
01
Identify Possible Rational Zeros
According to the Rational Zero Theorem, the possible rational zeros of a polynomial are the factors of the constant term (6) divided by the factors of the leading coefficient (8). The factors of 6 are \( \pm 1, \pm 2, \pm 3, \pm 6 \) and the factors of 8 are \( \pm 1, \pm 2, \pm 4, \pm 8 \). Thus, the possible rational zeros are: \( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{8}, \pm 2, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{4}, \pm 6 \).
02
Test Possible Zeros Using Synthetic Division
Using synthetic division, test each potential zero until one divides the polynomial evenly. Testing \(x = -\frac{1}{2}\), perform synthetic division:1. Write coefficients: \(8, 26, 39, 26, 6\).2. Bring down the 8.3. Multiply by \(-\frac{1}{2}\) and add to the next coefficient: - \(8 \cdot -\frac{1}{2} = -4 \), add to 26 = 22. - \(22 \cdot -\frac{1}{2} = -11 \), add to 39 = 28. - \(28 \cdot -\frac{1}{2} = -14 \), add to 26 = 12. - \(12 \cdot -\frac{1}{2} = -6 \), add to 6 = 0.The remainder is 0, so \(x = -\frac{1}{2}\) is a zero.
03
Divide Polynomial by \(x + \frac{1}{2}\)
Since \(x = -\frac{1}{2}\) is a zero, divide the original polynomial by \(x + \frac{1}{2}\). The quotient is \(8x^3 + 22x^2 + 28x + 12\).
04
Repeat Process for Remaining Polynomial
Now, apply the Rational Zero Theorem to \(8x^3 + 22x^2 + 28x + 12\) to identify and test other potential rational zeros, like -1 and -2. Testing these with synthetic division reveals:1. \(x = -1\) does not work (remainder not 0).2. Testing \(x = -2\) yields a remainder of 0, thus \(x = -2\) is a zero.
05
Factor and Solve Quadratic Equation
After confirming \(x = -2\) is a zero, divide \(8x^3 + 22x^2 + 28x + 12\) by \(x+2\), resulting in \(8x^2 + 6x + 6\). Solve \(8x^2 + 6x + 6 = 0\) using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\] where \(a = 8\), \(b = 6\), \(c = 6\).
06
Calculate Using the Quadratic Formula
Plug in values:\[x = \frac{-6 \pm \sqrt{6^2-4 \times 8 \times 6}}{2 \times 8}\] \[x = \frac{-6 \pm \sqrt{36-192}}{16}\] \[x = \frac{-6 \pm \sqrt{-156}}{16}\] Notice the discriminant is negative, indicating no further real zeros.
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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 divisor of the form \(x - c\). It's a quick method to test potential zeros of a polynomial without performing full polynomial long division. In synthetic division, you only use the coefficients from the polynomial and the value of \(c\), without explicitly writing the variables. Here's a simple way to understand the process:
- Write down the coefficients of the polynomial.
- Bring down the leading coefficient to the row below.
- Multiply this number by \(c\) (the value you are testing) and write the result under the next coefficient.
- Add this result to the next coefficient, bringing the sum down.
- Repeat the multiply and add process across all coefficients.
Polynomial Function
A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. For one variable \(x\), it has the general form:\[f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\]where:
- \(a_n, a_{n-1}, \ldots, a_0\) are constants, called coefficients.
- \(n\) is a non-negative integer, the degree of the polynomial, which is the highest power of \(x\) in the expression.
- \(a_n 鈮� 0\), ensuring the leading term is non-zero.
Quadratic Formula
The quadratic formula is a definitive method used to find the zeros of any quadratic equation, which is a polynomial of degree two. Quadratic equations take the form:\[ax^2 + bx + c = 0\]The quadratic formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here's how it works:
- The "\(a\), \(b\), \(c\)" in the formula are coefficients from the quadratic equation.
- "\(\pm\)" indicates that there could be two solutions: one involving addition and the other involving subtraction.
- The term \(b^2 - 4ac\) is called the discriminant. It helps determine the nature of the roots:
- If positive, there are two distinct real zeros.
- If zero, there's exactly one real zero.
- If negative, no real zeros exist, only complex ones.
Real Zeros
Real zeros of a polynomial function, also called roots or solutions, are the \(x\)-values where the polynomial evaluates to zero. These are the points where the graph of the polynomial intersects the \(x\)-axis. Finding the real zeros helps in understanding the behavior of the polynomial function. Several methods can be used to find real zeros:
- Rational Zero Theorem: Offers potential rational zeros which can be tested using methods like synthetic division.
- Factoring: When possible, factoring simplifies finding real zeros as roots of individual factors.
- Quadratic Formula: Solves quadratic polynomials (degree 2) to identify real zeros.
