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Magazine Chapter 3: Problem 57
Find all zeros of the polynomial. $$P(x)=x^{4}-6 x^{3}+13 x^{2}-24 x+36$$
Short Answer
Expert verified
Zeros are 1, 3, 1 + 3i, and 1 - 3i.
Step by step solution
01
Identify Polynomial Degree
The degree of the polynomial \(P(x) = x^4 - 6x^3 + 13x^2 - 24x + 36\) is 4. This indicates there are up to 4 real or complex roots to find.
02
Use the Rational Root Theorem
The Rational Root Theorem suggests that any rational root, \(p/q\), is a factor of the constant term (36) divided by a factor of the leading coefficient (1). Thus, potential rational roots are 卤1, 卤2, 卤3, 卤4, 卤6, 卤9, 卤12, 卤18, 卤36.
03
Test Potential Rational Roots with Synthetic Division
Use synthetic division to test possible roots starting from the smallest absolute values. Upon testing, \(x = 1\) satisfies the polynomial, as there is no remainder:
04
Perform Synthetic Division with x = 1
Divide the polynomial by \(x - 1\) using synthetic division. The quotient is \(x^3 - 5x^2 + 8x - 36\).
05
Find Zeros of the Quotient Polynomial
Repeat the process starting with \(x^3 - 5x^2 + 8x - 36\). By testing roots, \(x = 3\) satisfies the quotient polynomial.
06
Perform Synthetic Division with x = 3
Divide \(x^3 - 5x^2 + 8x - 36\) by \(x - 3\) using synthetic division. The quotient is \(x^2 - 2x + 12\).
07
Find Zeros of Quadratic Polynomial
Solve \(x^2 - 2x + 12\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1, b = -2, c = 12\). This gives complex roots \(x = 1 \pm 3i\).
08
Compile All Zeros
Combine all zeros from steps: \(x = 1\), \(x = 3\), and complex roots \(x = 1 + 3i, x = 1 - 3i\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Rational Root Theorem
The Rational Root Theorem is a powerful tool in algebra that helps in finding potential rational zeros of a polynomial. Here's how it works:
The theorem states that if a polynomial has a rational root \(\frac{p}{q}\), then \(p\) must be a factor of the constant term and \(q\) must be a factor of the leading coefficient. For the polynomial \(P(x) = x^4 - 6x^3 + 13x^2 - 24x + 36\), the constant term is 36, and the leading coefficient is 1.
The theorem states that if a polynomial has a rational root \(\frac{p}{q}\), then \(p\) must be a factor of the constant term and \(q\) must be a factor of the leading coefficient. For the polynomial \(P(x) = x^4 - 6x^3 + 13x^2 - 24x + 36\), the constant term is 36, and the leading coefficient is 1.
- Factors of 36: 卤1, 卤2, 卤3, 卤4, 卤6, 卤9, 卤12, 卤18, 卤36
- Factors of 1: 卤1
Synthetic Division
Synthetic division is a simplified method of dividing a polynomial by a linear binomial of the form \(x - c\). It is particularly helpful for checking the roots of a polynomial quickly.
Here's how synthetic division is performed:
1. Write down the coefficients of the polynomial.2. Identify the root you're testing (let's say \(x = 1\)).3. Perform the synthetic division process, which involves multiplying and adding, to see if the remainder is zero.
If you get a remainder of zero, then \(x = 1\) is a root of the polynomial. In our case, this step was used twice: first finding \(x = 1\), and then \(x = 3\), confirming that these values are indeed the roots.
Here's how synthetic division is performed:
1. Write down the coefficients of the polynomial.2. Identify the root you're testing (let's say \(x = 1\)).3. Perform the synthetic division process, which involves multiplying and adding, to see if the remainder is zero.
If you get a remainder of zero, then \(x = 1\) is a root of the polynomial. In our case, this step was used twice: first finding \(x = 1\), and then \(x = 3\), confirming that these values are indeed the roots.
Complex Roots
Complex roots arise when the discriminant of a quadratic equation (inside the square root of the quadratic formula) is negative. These are numbers that include a real part and an imaginary part.
For the quadratic \(x^2 - 2x + 12\), we used the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
With values \(a = 1\), \(b = -2\), and \(c = 12\), the discriminant \(b^2 - 4ac\) evaluates to \(-44\), which is negative. This indicates complex roots. The roots are calculated as:
\[ x = 1 \pm 3i \]
Here, "i" represents the imaginary unit, and these roots are parts of the solution we compiled at the end.
For the quadratic \(x^2 - 2x + 12\), we used the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
With values \(a = 1\), \(b = -2\), and \(c = 12\), the discriminant \(b^2 - 4ac\) evaluates to \(-44\), which is negative. This indicates complex roots. The roots are calculated as:
\[ x = 1 \pm 3i \]
Here, "i" represents the imaginary unit, and these roots are parts of the solution we compiled at the end.
Quadratic Formula
The quadratic formula is a universal method used to find the roots of a quadratic equation \(ax^2 + bx + c = 0\). It can be used regardless of the type of roots: real or complex.
The formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula takes into account the coefficients \(a\), \(b\), and \(c\) from the quadratic equation and calculates the roots based on the discriminant \(b^2 - 4ac\).
The formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula takes into account the coefficients \(a\), \(b\), and \(c\) from the quadratic equation and calculates the roots based on the discriminant \(b^2 - 4ac\).
- If the discriminant is positive, the roots are real and different.
- If it is zero, roots are real and the same.
- If it's negative, the roots are complex.
