Problem 14 - College Algebra with Modeling and Visualization

Question

Let \(a_{n}\) be the leading coefficient. (a) Find the complete factored form of a polynomial with real coefficients \(f(x)\) that satisfies the conditions. (b) Express \(f(x)\) in expanded form. Degree \(2 ; a_{n}=-5 ; \quad\) zeros \(1+i\) and \(1-i\)

Step-by-Step Solution

Answer

The complete factored form is \(-5((x - 1)^2 + 1)\), and the expanded form is \(-5x^2 + 10x - 10\).

Step 1: Use Zeros to Build Factors

The polynomial has complex zeros \(1+i\) and \(1-i\). Since these are conjugates and the polynomial must have real coefficients, they form a quadratic factor. Use each zero to create linear factors: \((x - (1+i))\) and \((x - (1-i))\).

Step 2: Multiply Complex Conjugate Factors

Multiply the factors from Step 1. Start with:\[ (x - (1+i))(x - (1-i)) = (x - 1 - i)(x - 1 + i) \]This is a difference of squares, which simplifies to:\[ (x - 1)^2 - (i)^2 = (x - 1)^2 - (-1) = (x-1)^2 + 1 \].

Step 3: Expand the Quadratic Expression

Expand \((x - 1)^2 + 1\):First, expand \((x - 1)^2\):\[ (x - 1)^2 = x^2 - 2x + 1 \]Adding 1 gives:\[ x^2 - 2x + 1 + 1 = x^2 - 2x + 2 \].

Step 4: Include Leading Coefficient

The leading coefficient given is \(-5\). Multiply the quadratic expression from Step 3 by \(-5\) to include it in the polynomial:\[ f(x) = -5(x^2 - 2x + 2) \].

Step 5: Expand to General Form

Distribute \(-5\) across \(x^2 - 2x + 2\):\[ f(x) = -5x^2 + 10x - 10 \].

Key Concepts

Complex Zeros

Complex zeros are values of a polynomial that involve imaginary numbers. When dealing with real coefficient polynomials, complex zeros will always appear in conjugate pairs. This means if you have a zero like \(1+i\), its conjugate, \(1-i\), will also be a zero. This pairing is essential because it helps ensure that the polynomial remains real, even when complex numbers are involved.
Complex zeros are a natural extension of real zeros, and they allow us to find roots of all polynomials using the fundamental theorem of algebra.
When constructing factors from complex zeros, each zero translates into a linear factor. Therefore, from zeros \(1+i\) and \(1-i\), we create the factors \((x-(1+i))\) and \((x-(1-i))\). By multiplying these linear factors, we keep the polynomial real while still capturing the behavior of the complex zeros.

Quadratic Factors

Quadratic factors result from multiplying pairs of complex conjugate factors. A quadratic factor is a polynomial of degree two, usually in the form \(ax^2 + bx + c\).
In our example, we create a quadratic factor by multiplying the linear factors associated with the conjugate zeros \((x-(1+i))\) and \((x-(1-i))\).
This multiplication yields a quadratic factor:

• Start with the difference of squares identity:
• \( (x-1-i)(x-1+i) = (x-1)^2 - i^2 = (x-1)^2 + 1 \)
• This simplifies into the quadratic factor \((x-1)^2 + 1 = x^2 - 2x + 2\).

Quadratic factors formed from complex conjugates give us a useful tool for keeping the coefficients of polynomials real.

Real Coefficients

Polynomials with real coefficients have numbers in the polynomial that are all real, meaning they don't include any imaginary numbers. This requirement is crucial in most practical applications, as real numbers are more intuitive to understand and use.
When a polynomial is described to have real coefficients, it naturally handles complex zeros by pairing them with their conjugates:

• A conjugate pair ensures that any imaginary components cancel out when multiplied together.
• This cancellation leaves a polynomial with purely real numbers.

In our case, by including complex zeros \(1+i\) and \(1-i\) as factors, the polynomial ensures the coefficients are real, primarily through the formation of the quadratic factor \(x^2 - 2x + 2\). This behavior guarantees that no imaginary numbers appear in the polynomial, maintaining real coefficients throughout.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient in front of the highest degree term. It's important because it influences the end behavior of the polynomial graph and the stretch or compression of the polynomial function.
In our example, the given leading coefficient is \(-5\). This affects the quadratic factor \(x^2 - 2x + 2\) by multiplying it entirely to form:

• \(f(x) = -5(x^2 - 2x + 2)\)
• Which expands to: \(f(x) = -5x^2 + 10x - 10\)

The leading coefficient of \(-5\) not only amplifies the polynomial but also reverses its direction, causing the parabola (if graphed) to open downwards. Leading coefficients are vital for determining the nature and behavior of polynomial functions.