Problem 62 - Algebra and Trigonometry

Question

Find all zeros of the polynomial. \(P(x)=x^{5}-2 x^{4}+2 x^{3}-4 x^{2}+x-2\)

Step-by-Step Solution

Answer

The zeros are \(x = 2, x = i, x = -i\).

Step 1: Attempt Rational Root Theorem

To find all zeros of the polynomial, we can use the Rational Root Theorem, which suggests that any rational root of the polynomial must be a factor of the constant term (-2) divided by a factor of the leading coefficient (1). So, we look at possible rational roots: ±1, ±2.

Step 2: Test Potential Rational Roots

Substitute the potential rational roots into the polynomial to check which ones are actual roots.\- For \(x = 1\):\[P(1) = 1^5 - 2*1^4 + 2*1^3 - 4*1^2 + 1 - 2 = -4\]\- For \(x = -1\):\[P(-1) = (-1)^5 - 2(-1)^4 + 2(-1)^3 - 4(-1)^2 + (-1) - 2 = -12\] \- For \(x = 2\):\[P(2) = 2^5 - 2*2^4 + 2*2^3 - 4*2^2 + 2 - 2 = 0\]\- For \(x = -2\):\[P(-2) = (-2)^5 - 2(-2)^4 + 2(-2)^3 - 4(-2)^2 - 2 - 2 = -100\]\Thus, \(x = 2\) is a root.

Step 3: Polynomial Division

Since \(x = 2\) is a root, divide the polynomial by \(x - 2\) using synthetic division or polynomial long division to reduce the polynomial's degree.

Step 4: Perform Synthetic Division

Synthetic division with \(x = 2\):\Write down the coefficients: [1, -2, 2, -4, 1, -2].\1. Bring down the leading coefficient: 1.\2. Multiply 1 by 2 (the root) and add to the next coefficient: -2 + 2 = 0.\3. Multiply 0 by 2 and add to the next coefficient: 2 + 0 = 2.\4. Multiply 2 by 2 and add to the next coefficient: -4 + 4 = 0.\5. Multiply 0 by 2 and add to the next coefficient: 1 + 0 = 1.\6. Multiply 1 by 2 and add to the next coefficient: -2 + 2 = 0.\The quotient is \(x^4 + 0x^3 + 2x^2 + 0x + 1\).

Step 5: Find Zeros of the Quadratic

The quotient is \(x^4 + 2x^2 + 1\). Since it is not factored easily or reducible using direct factoring methods, try substituting \(x^2 = y\), making it \(y^2 + 2y + 1 = 0\). Factor the quadratic expression as a perfect square: \((y + 1)^2 = 0\), so \(y = -1\). Thus, \(x^2 = -1\).

Step 6: Solve for x

Since \(x^2 = -1\), solving gives \(x = i\) and \(x = -i\), where \(i\) is the imaginary unit, representing the square root of -1.

Step 7: Collect All Zeros

The zeros of the polynomial \(P(x) = x^5 - 2x^4 + 2x^3 - 4x^2 + x - 2\) are \(x = 2, x = i, x = -i\).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a useful tool in algebra that helps us find the possible rational roots of a polynomial equation. It states that for a polynomial with integer coefficients, any rational solution, expressed as a fraction \( \frac{p}{q} \), must have \( p \) as a factor of the constant term and \( q \) as a factor of the leading coefficient. In simpler terms, if a polynomial has a rational root, it is one of the fractions formed by dividing the factors of the constant term by the factors of the leading coefficient. This method helps us limit the number of potential roots we need to test.For the polynomial \( P(x) = x^5 - 2x^4 + 2x^3 - 4x^2 + x - 2 \), the Rational Root Theorem suggests that the potential rational roots are ±1 and ±2, since -2 is the constant term, and 1 is the leading coefficient. By substituting these values into the polynomial, we can identify whether any are actual roots. As shown in the solution, only \( x = 2 \) is a root.

synthetic division

Synthetic division is a simplified form of polynomial division, typically used when dividing by a linear factor, such as \( x - a \). It's a more straightforward process compared to long division, as it involves less writing and fewer calculations. Synthetic division is particularly helpful when verifying possible roots of a polynomial, as it quickly reduces the polynomial's degree.To use synthetic division in the original exercise, we consider the root \( x = 2 \) of the polynomial \( P(x) = x^5 - 2x^4 + 2x^3 - 4x^2 + x - 2 \). Write down the coefficients: [1, -2, 2, -4, 1, -2]. Next, perform the division.

• Bring down the leading coefficient, 1.
• Multiply it by the root (2) and add this product to the next coefficient, -2, resulting in 0.
• Repeat this process for remaining coefficients.

The synthetic division shows that the quotient of the division is \( x^4 + 2x^2 + 1 \), confirming that \( x = 2 \) is a root.

complex numbers

Complex numbers are an essential part of algebra, especially when dealing with polynomials that do not have real roots. They are expressed in the form \( a + bi \), where \( i \) is the imaginary unit, defined as \( i^2 = -1 \). Complex numbers expand the real number system and are crucial when working with quadratic equations that have negative discriminants.In our exercise involving the polynomial \( P(x) = x^5 - 2x^4 + 2x^3 - 4x^2 + x - 2 \), after synthetic division, we derive the equation \( x^4 + 2x^2 + 1 = 0 \). Factoring or substituting shows that the quadratic \( x^2 = -1 \) needs to be solved. Here, the solutions are \( x = i \) and \( x = -i \), both complex numbers. These complex solutions demonstrate the completeness of the polynomial roots and are invaluable for solving equations where real numbers fall short. This highlights the importance of understanding complex numbers in algebra.