Problem 24 - 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. $$-2+i,-2-i, 3, \text { and }-3$$

Step-by-Step Solution

Answer

The polynomial is \(P(x) = x^4 + 4x^3 - 4x^2 - 36x - 45\).

Step 1: Identify Zeros and Their Conjugates

The given zeros of the polynomial are complex and real: \(-2+i\), \(-2-i\), 3, and \(-3\). Since complex roots occur in conjugate pairs in polynomials with real coefficients, \(-2+i\) and \(-2-i\) are a conjugate pair and contribute to the polynomial.

Step 2: Form Factor Pairs from Zeros

Create factors of the polynomial from each zero: For zero \(-2+i\), the factor is \((x - (-2+i)) = (x + 2 - i)\) and for \(-2-i\), it is \((x - (-2-i)) = (x + 2 + i)\). For zero 3, the factor is \((x - 3)\), and for zero -3, it is \((x + 3)\).

Step 3: Form Polynomial from Complex Zeros

Multiply the complex conjugate pairs to simplify: \ \((x + 2 - i)(x + 2 + i) = (x + 2)^2 - i^2 = (x + 2)^2 + 1 = x^2 + 4x + 5\) (since \(i^2 = -1\)).

Step 4: Combine All Factors into a Polynomial

Now multiply all factors together: \ \((x^2 + 4x + 5)(x - 3)(x + 3)\). Use the difference of squares: \ \((x^2 - 3^2) = (x^2 - 9)\).

Step 5: Expand the Polynomial

Expand the polynomial by multiplying: \ \((x^2 + 4x + 5)(x^2 - 9)\). Use distributive property: \ First, \((x^2 + 4x + 5) \cdot x^2 = x^4 + 4x^3 + 5x^2\), \ Next, \((x^2 + 4x + 5) \cdot (-9) = -9x^2 - 36x - 45\). \ Then, combine: \ \(x^4 + 4x^3 + 5x^2 - 9x^2 - 36x - 45 = x^4 + 4x^3 - 4x^2 - 36x - 45\).

Key Concepts

Complex Conjugates

In polynomial functions with real coefficients, complex roots always appear in pairs known as complex conjugates. This means that if a polynomial has a complex zero, such as -2+i, its conjugate, -2-i, will also be a zero. This pairing is essential because when these two roots are multiplied, the imaginary parts cancel out, ensuring the polynomial retains its real coefficients.

• The conjugate of a complex number a+bi is a-bi.
• By forming pairs like (x + 2 - i)(x + 2 + i), the result is a real quadratic factor.

These pairs simplify the complexity of working with polynomials, allowing us to focus on constructing real-coefficient polynomials.

Real Coefficients

Real coefficients in polynomials imply that each term of the polynomial is a real number. This is a common requirement, making them applicable to a wide range of real-world problems. When working with complex zeros, you need to ensure that they appear in conjugate pairs so that they combine to produce real factors. This is seen in the step where (x + 2 - i) and (x + 2 + i) are multiplied, simplifying to x^2 + 4x + 5. Here, each term is real.

• When multiplying complex conjugates, the imaginary components cancel out.
• Ensure that polynomial equations maintain real coefficients by pairing conjugate roots.

This technique contributes to creating well-structured polynomials ready for further algebraic operations.

Difference of Squares

The difference of squares is a handy algebraic identity that can simplify certain polynomial expressions. It is defined as (a^2 - b^2) = (a - b)(a + b). This identity is useful when dealing with polynomials that involve factors like (x - 3)(x + 3). Using the difference of squares allows us to quickly identify and simplify expressions obtained after pairing or expanding terms.

• The expression (x - 3)(x + 3) becomes (x^2 - 9), which is straightforward.
• This approach reduces complexity in polynomial expansion.

Recognizing patterns like the difference of squares can streamline polynomial manipulations and expansions.

Expanding Polynomial

Expanding a polynomial involves multiplying the individual factors to form a single polynomial expression. This process transforms products of binomials and other expressions into a standard polynomial form. For example, after forming the simplified factors, the polynomial (x^2 + 4x + 5)(x^2 - 9) is expanded using the distributive property. The key steps involve:

• Multiplying each term in the first polynomial by each term in the second polynomial.
• Using the distributive property to ensure all terms are accounted for and properly combined.
• Simplifying by combining like terms to form the final polynomial: x^4 + 4x^3 - 4x^2 - 36x - 45.

This meticulous approach helps to ensure that the polynomial is expressed clearly and accurately, ready for analysis or further use.