Problem 28 - A Graphical Approach to College Algebra

Question

Find a polynomial function \(P(x)\) having leading coefficient \(1,\) least possible degree, real coefficients, and the given zeros. $$1+2 i \text { and } 2 \text { (multiplicity } 2)$$

Step-by-Step Solution

Answer

The polynomial is \(x^4 - 6x^3 + 9x^2 + 4x - 12\).

Step 1: Identify the Zeros

The given zeros of the polynomial are \(1 + 2i\) and \(2\) with multiplicity 2. Since the polynomial must have real coefficients, the complex conjugate of \(1 + 2i\), which is \(1 - 2i\), should also be a zero.

Step 2: Construct Linear Factors

Each zero corresponds to a linear factor of the polynomial. For the zero \(1 + 2i\), the factor is \((x - (1 + 2i))\), and for the zero \(1 - 2i\), the factor is \((x - (1 - 2i))\). The zero with multiplicity, 2, corresponds to \((x - 2)^2\). Therefore, the factors of the polynomial are \((x - (1+2i))(x - (1-2i))\) and \((x-2)^2\).

Step 3: Multiply the Complex Conjugate Factors

Multiply the complex conjugate factors to form a quadratic polynomial with real coefficients. This is done by expanding \[(x - (1 + 2i))(x - (1 - 2i)) = (x - 1 - 2i)(x - 1 + 2i) = ((x - 1)^2 - (2i)^2) = (x - 1)^2 - 4\].Now, expand \((x - 1)^2\) to get:\[(x - 1)^2 = x^2 - 2x + 1\].Substitute back:\[x^2 - 2x + 1 - 4 = x^2 - 2x - 3\].

Step 4: Multiply with the Factor from the Multiple Root

Now multiply the result from Step 3 by the factor corresponding to the root with multiplicity, \((x - 2)^2\), which expands to \[(x - 2)^2 = x^2 - 4x + 4\]. The polynomial can be found by multiplying \((x^2 - 2x - 3)\) by \((x^2 - 4x + 4)\).

Step 5: Polynomial Expansion

Perform polynomial multiplication:\[(x^2 - 2x - 3)(x^2 - 4x + 4)\].Distribute all terms:-\((x^2)\cdot(x^2 - 4x + 4) = x^4 - 4x^3 + 4x^2\).-\((-2x)\cdot(x^2 - 4x + 4) = -2x^3 + 8x^2 - 8x\).-\((-3)\cdot(x^2 - 4x + 4) = -3x^2 + 12x - 12\).Sum all these terms:\[x^4 - 4x^3 + 4x^2 - 2x^3 + 8x^2 - 8x - 3x^2 + 12x - 12\].

Step 6: Combine Like Terms

Combine the same degree terms from the expansion:- \(x^4\) term: \(x^4\)- \(x^3\) terms: \(-4x^3 - 2x^3 = -6x^3\)- \(x^2\) terms: \(4x^2 + 8x^2 - 3x^2 = 9x^2\)- \(x\) terms: \(-8x + 12x = 4x\)- Constant term: \(-12\)The polynomial is \[x^4 - 6x^3 + 9x^2 + 4x - 12\].

Step 7: Verify the Leading Coefficient

Check that the leading coefficient of the polynomial \(x^4 - 6x^3 + 9x^2 + 4x - 12\) is 1, which matches the requirement in the problem statement.

Key Concepts

Complex Conjugates

In the world of polynomial functions, complex numbers often come in pairs known as complex conjugates. A complex conjugate is obtained by changing the sign of the imaginary part of a complex number. For example, given a complex number like \(1 + 2i\), its complex conjugate is \(1 - 2i\).

• When working with polynomials that have real coefficients, it is essential that complex roots appear in conjugate pairs. This ensures that when a polynomial is expanded, the imaginary parts cancel out, resulting in a polynomial with real coefficients.

If we have a root \(1 + 2i\), its conjugate \(1 - 2i\) must also be a root to keep the polynomial real. In constructing polynomials, always remember to include both a complex number and its complex conjugate if the coefficients of your desired polynomial must be real.

Multiplicity of Roots

Multiplicity refers to how many times a particular root appears in a polynomial. For instance, if the root \(2\) has multiplicity 2, it means that \((x-2)\) is squared in the factorized form of the polynomial.

• Roots with higher multiplicity imply that the polynomial will "touch" its axis at the root but not cross it (for even multiplicity) or will flatten out at that point (for odd multiplicity).
• Including the multiplicity of roots in polynomial expressions reflects crucial information about the behavior and nature of the graph of these functions.

In the given example, having a root \(2\) with multiplicity 2 indicates that the polynomial has the factor \((x-2)^2\). This leads to a polynomial that dips or "flattens" at \(x = 2\) but not passing through. A polynomial with root multiplicities truly captures the essence of its graph's behavior.

Polynomial Expansion

Polynomial expansion involves transforming a product of factors into a sum or difference of terms. When dealing with polynomials, expansion is the process where each term in each factor is distributed across terms of other factors. This results in a single polynomial expression.

• A typical procedure starts by multiplying linear factors together, like \((x-a)(x-b)\).
• The power of expansion in polynomials allows for simplification and combining like terms, which manifests in the polynomial's simplest form.This also makes further calculus problems (such as finding derivatives) more manageable.

When we expand a polynomial, namely from something like \((x-(1+2i))(x-(1-2i))(x-2)^2\), we use distributive properties to simplify it into a neat polynomial. This involves handling the imaginary parts first (to combine like terms effectively) and concluding with real number multiplication resulting in a real coefficients polynomial.

Real Coefficients

Polynomials can have coefficients that are real numbers. Real coefficients are essential in specific scenarios, especially when attempting to ensure that the polynomial's graph is tangible and observable.

• A polynomial with real coefficients will not have terms that result in any unexplainable imaginary graph segments.
• Real coefficients ensure the polynomial functions over real number fields, which is crucial for real-world applications where complex coefficients don't make sense.

In the process of creating our polynomial, the requirement was for the resulting function to have real coefficients. This importance led to the inclusion of complex conjugates in our polynomial roots. As these conjugates cancel each other's imaginary parts out, the polynomial ends up satisfying the condition of having real coefficients only. This makes it applicable in numerous typical geometric and algebraic applications.