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Magazine Chapter 3: Problem 31
Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=3 x^{4}-10 x^{3}-9 x^{2}+40 x-12$$
Short Answer
Expert verified
The rational zeros are 2 and -\( \frac{1}{3} \). The factored form of \(P(x)\) is \(3(x - 2)(3x + 1)(x^2 - \frac{5}{3}x + 2)\).
Step by step solution
01
List Possible Rational Zeros
According to the Rational Root Theorem, any rational solution, expressed as \( \frac{p}{q} \), where \( p \) is a factor of the constant term \(-12\) and \( q \) is a factor of the leading coefficient \(3\). The factors of \(-12\) are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \), and the factors of \(3\) are \( \pm 1, \pm 3 \). Thus, the possible rational roots are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3} \).
02
Test Rational Zeros Using Synthetic Division
We use synthetic division to test these potential zeros. We'll start with \(1\). Perform synthetic division on \(P(x)\) with \(x = 1\). This does not yield a zero remainder. Repeat this process for other potential roots such as \(-1, 2, -2, 3, -3\), and \(\frac{1}{3}\) to find which yield zero remainders.
03
Confirm Rational Zeros
Upon testing, we find that \(x = 2\) yields a zero remainder using synthetic division. Thus, \(x = 2\) is a zero of \(P(x)\). Dividing \(P(x)\) by \(x - 2\) gives \(3x^3 - 4x^2 - 17x + 6\).
04
Repeat Process for Reduced Polynomial
Apply the same process to \(3x^3 - 4x^2 - 17x + 6\). Testing possible rational zeros, we find \(x = -\frac{1}{3}\) is another zero, leading to division. This gives \(3x^2 - 5x + 6\) after removing \(x + \frac{1}{3}\) from the polynomial.
05
Solve Quadratic Polynomial
Solve \(3x^2 - 5x + 6\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 3\), \(b = -5\), \(c = 6\). Calculating it gives complex roots \(x = \frac{5 \pm i\sqrt{-47}}{6}\), indicating no further rational roots.
06
Write the Polynomial in Factored Form
Thus, the polynomial \(P(x)\) can be factored into \(3(x - 2)(x + \frac{1}{3})\cdot(3x^2 - 5x + 6)\). Further simplification gives \(P(x) = 3(x - 2)(3x + 1)(x^2 - \frac{5}{3}x + 2)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial factorization
Polynomial factorization is the process of expressing a polynomial as a product of its factors. This helps in simplifying complex polynomial expressions and finding the roots or solutions. To factor a polynomial like \( P(x) = 3x^4 - 10x^3 - 9x^2 + 40x - 12 \), we aim to break it down into simpler polynomial factors.
- **Identify Possible Rational Zeros:** Use the Rational Root Theorem, which states that if a polynomial has a rational root \( \frac{p}{q} \), then \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient.
- **Test for Possible Zeros:** Use a method like synthetic division to test each possibility until we find an actual zero or root of the polynomial.
- **Divide and Conquer:** Once a zero is found, divide the polynomial by the corresponding factor, effectively reducing the degree of the polynomial.
Synthetic division
Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form \( x - c \). This method is particularly useful when testing potential rational roots identified by the Rational Root Theorem. Synthetic division simplifies the evaluation process significantly compared to traditional long division.
- **Setup:** Write down the coefficients of the polynomial you are dividing. For example, using \( P(x) = 3x^4 - 10x^3 - 9x^2 + 40x - 12 \), you'd list \( 3, -10, -9, 40, -12 \).
- **Test a Root:** Choose a possible root (like \( x = 2 \)) and use it to perform synthetic division. You substitute this value for \( c \) and perform the operations.
- **Operation:** Multiply and add through each step, carrying down values. If you end with a zero remainder, the chosen \( c \) is a root, and the polynomial is divisible by \( x - c \).
Quadratic formula
The quadratic formula provides a straightforward way to find the roots of a quadratic equation of the form \( ax^2 + bx + c = 0 \). This formula is particularly useful when the quadratic does not factor easily or when dealing with complex numbers. The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
- **Coefficients:** Identify the coefficients \( a \), \( b \), and \( c \) from the quadratic equation resulting from the division of the polynomial.
- **Discriminant:** Calculate \( b^2 - 4ac \), which determines the nature of the roots. A positive value indicates real roots; a negative value, complex roots.
- **Solve:** Substitute the values of \( a \), \( b \), and \( c \) into the quadratic formula to find the roots.
