Problem 47 - Algebra 2

Question

Given a function and one of its zeros, find all of the zeros of the function. \(h(x)=x^{3}-11 x+20 ; 2+i\)

Step-by-Step Solution

Answer

The zeros of the function are \(0, 2+i, 2-i\).

Step 1: Identify the Zero and Its Conjugate

The given complex number zero is \(2+i\). Since the coefficients of the polynomial are real, the complex conjugate \(2-i\) is also a zero of the function. So far, we have zeros \(2+i\) and \(2-i\).

Step 2: Form a Quadratic from Complex Zeros

Since \(2+i\) and \(2-i\) are conjugate zeros, they can form a quadratic factor of the polynomial. Multiply the conjugate pair: \((x-(2+i))(x-(2-i)) = (x-2-i)(x-2+i) = (x-2)^2-i^2 = (x-2)^2+1\). Simplify to get \(x^2-4x+5\).

Step 3: Perform Polynomial Division

Divide the original polynomial \(h(x) = x^3 - 11x + 20\) by \(x^2 - 4x + 5\) using polynomial long division. This will find the remaining linear factor.

Step 4: Complete the Division

Perform the polynomial division:1. Divide the leading term \(x^3\) by \(x^2\), result is \(x\).2. Multiply \(x\) by \(x^2 - 4x + 5\) which is \(x^3 - 4x^2 + 5x\).3. Subtract \(x^3 - 4x^2 + 5x\) from \(x^3 - 11x + 20\) giving \(4x^2 - 16x + 20\).4. Repeat with new leading term \(4x^2\): Divide \(4x^2\) by \(x^2\), result is \(4\). Multiply \(4\) by \(x^2 - 4x + 5\) which is \(4x^2 - 16x + 20\).5. Subtraction gives remainder zero, confirming the division.

Step 5: Identify All Zeros

The quotient from the division is \(x\), corresponding to the zero \(x = 0\). Considering all zeros, they are \(2+i\), \(2-i\), and \(0\).

Key Concepts

Complex Conjugates

In the fascinating world of complex numbers, each complex number has a twin known as its complex conjugate. If you have a complex number written in the form \(a + bi\), its conjugate would be \(a - bi\). These pairs are particularly important when working with polynomials that have real coefficients. Here's why

• When a polynomial has real coefficients and a complex number \(2+i\) is a zero, its complex conjugate \(2-i\) is also automatically a zero.
• In practice, knowing one zero allows you to immediately identify its conjugate as another zero.

Therefore, if you are given \(2+i\) as a zero of a polynomial, you do not need to find the second zero from scratch. Just use the properties of complex numbers to quickly gain its conjugate's info as well.
This leads us seamlessly into our next concept, which is forming a quadratic factor using these complex conjugates.

Polynomial Division

Polynomial division is a really useful tool when dealing with the zeros of polynomials. Once you're aware of a quadratic factor formed from complex conjugates, you can use it to simplify the polynomial.
Here's a simple overview of how polynomial division works:

• Start by dividing the highest degree term of your polynomial by the highest degree term of the factor you're using. This gives you the first term of the quotient.
• Multiply this term by the entire divisor (the quadratic factor), and write the result under the terms of the polynomial. Subtract these from the polynomial.
• You get a new polynomial, and you repeat the process with the next highest power term.

This systematic process allows for effective determination of remaining factors and zeros of the polynomial. In our example, dividing \(x^3 - 11x + 20\) by \(x^2 - 4x + 5\) reveals another zero as well as confirms the factorization.

Quadratic Factorization

Quadratic factorization is another critical step in solving polynomial equations. Once you have the zeros of a polynomial, particularly when they include complex numbers and their conjugates, you can leverage these to create a quadratic factor.
Consider how this operates here:

• Start with the complex conjugate zero pair, for example, \(2+i\) and \(2-i\).
• Plug these into the expression \((x-(2+i))(x-(2-i))\).
• Recognize that multiplying out these binomials involves using the difference of squares formula which simplifies to \((x-2)^2-i^2\).
• \(-i^2\) is \(+1\), so further simplification lands us at our quadratic factor \(x^2 - 4x + 5\).

Applying this factorization technique allows for efficient and effective solving of polynomials, revealing zeros that might otherwise remain elusive.