Problem 11 - Algebra and Trigonometry Real Mathematics, Real People

Question

Confirm that the function has the indicated zeros. $$f(x)=3 x^{4}-48 ;-2,2,-2 i, 2 i$$

Step-by-Step Solution

Answer

To conclude, substituting each potential root (-2, 2, -2i, 2i) into the function \(f(x) = 3x^{4} - 48\) and receiving a result of zero confirms that each of these values are roots of the function.

Step 1: Check -2 as root

Begin by substituting -2 into the function: \(f(-2) = 3(-2)^{4} - 48 = 3*16 - 48 = 48 - 48 = 0\). Since the outcome equals zero, -2 is a valid root of the function.

Step 2: Check 2 as root

Proceed by substituting 2 into the function: \(f(2) = 3(2)^{4} - 48 = 3*16 - 48 = 48 - 48 = 0\). Since the result is zero, 2 is a valid root of the function.

Step 3: Check -2i as root

Substitute -2i into the function: \(f(-2i) = 3(-2i)^{4} - 48 = 3*16 - 48 = 48 - 48 = 0\). As the calculation equals zero, -2i is a confirmed root.

Step 4: Check 2i as root

Finally, substitute 2i: \(f(2i) = 3(2i)^{4} - 48 = 3*16 - 48 = 48 - 48 = 0\). As the result is equals to zero, 2i is a validated root.

Key Concepts

Polynomial Roots

Understanding polynomial roots is fundamental in algebra. A root of a polynomial is a number that, when substituted for the variable, makes the polynomial equal to zero. In the given exercise, we have a polynomial function, f(x) = 3x^4 - 48, and we're asked to validate whether the numbers -2, 2, -2i, and 2i are indeed roots of this function.

By substituting these numbers into the function and finding that they yield a result of zero, we affirm that they are roots of the polynomial. For real numbers like -2 and 2, confirmation is more straightforward, as they typically represent points where the graph of the polynomial intersects the x-axis. However, when dealing with complex numbers like -2i and 2i, the concept might seem more abstract because these roots correspond to intersections in a two-dimensional complex plane, rather than on a one-dimensional number line.

Complex Zeros

Complex zeros, like the ones found in the exercise with -2i and 2i, are roots of polynomials that have an imaginary component. While real zeros can be easily visualized on the graph of a function, complex zeros cannot be represented on the standard Cartesian coordinate system. Instead, they exist in a complex plane where the horizontal axis is the real part and the vertical axis is the imaginary part.

The occurrence of complex zeros always follows a pattern—they appear in conjugate pairs which means if (a + bi) is a zero then its conjugate (a - bi) is also a zero. This pattern is shown in the exercise with -2i and 2i being conjugates of each other. Discovering complex zeros is essential when solving polynomial equations, as it provides a complete set of solutions when combined with the real zeros.

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every non-zero, single-variable, degree-n polynomial with complex coefficients has, counted with multiplicity, exactly n roots in the complex number set. This theorem guarantees that the polynomial function provided in the exercise, which is of the fourth degree, will have exactly four roots.

In this case, we have two real roots (-2 and 2) and two complex roots (-2i and 2i). This adheres to the theorem, providing us with four distinct roots that validate the solution to the exercise. Understanding this theorem helps students realize that the search for roots is not in vain; there is a precise number of solutions to be found for any given polynomial.

Synthetic Division

Synthetic division is a shortcut method of polynomial division, particularly when dividing by a linear factor - and much faster than the long division approach. Though not directly used in the given exercise, it's a powerful tool to understand since it can be employed to simplify polynomials and find roots efficiently. It works by decomposing the polynomial into smaller, more manageable pieces.

For example, if we were to have a hypothetical root 'r', we could use synthetic division to divide the polynomial by (x - r), helping us see if 'r' is indeed a root, based on whether the remainder is zero. Synthetic division is particularly useful when paired with the Rational Root Theorem or to verify potential roots found by other means, such as graphing or using the quadratic formula.