Question

Write a third or fourth degree polynomial function that has zeros at \(\pm \frac{3}{4}\). Justify your answer.

Short answer

The third degree polynomial with zeros at \( \pm \frac{3}{4} \) can be \(p(x) = (x^2 - \frac{9}{16}) \cdot (x - \frac{3}{4})\), and the fourth degree polynomial can be \(p(x) = (x^2 - \frac{9}{16})^2\).

Step-by-step solution

Form the factors

Start with the roots provided, and write them as factors of a polynomial. For the given zeros \( \pm \frac{3}{4} \), we would get two factors, \( (x - \frac{3}{4}) \) and \( (x + \frac{3}{4}) \).

Expand the equation for the zeros

Next, write the polynomial derived from these factors. The product of these two factors gives us the polynomial \(x^2 - \left(\frac{3}{4}\right)^2\), which simplifies to \(x^2 - \frac{9}{16}\).

Form third and fourth degree polynomials

To form a third degree polynomial, we simply add a multiplicity to one of the factors to have it repeated. The polynomial \(p(x) = (x^2 - \frac{9}{16}) \cdot (x - \frac{3}{4})\) is a third degree polynomial with zeros at \( \pm \frac{3}{4} \), with added multiplicity at \(- \frac{3}{4}\). For a fourth degree polynomial, add more multiplicity, leading to \(p(x) = (x^2 - \frac{9}{16})^2\), which is a fourth degree polynomial with zeros at \( \pm \frac{3}{4}\) each repeated twice.

Key concepts

Zeros of a Polynomial

Understanding the zeros of a polynomial is fundamental when studying polynomial functions. These are the values of 'x' for which the polynomial equation equals zero. They are also commonly known as the 'roots' or 'solutions' of the equation.

For example, if we have a polynomial equation such as \(f(x) = (x - a)(x - b)\), the zeros of this polynomial would be 'a' and 'b', because setting \(x\) equal to \(a\) or \(b\) would make the entire equation equal to zero. In the context of our original problem, the zeros were given as \(\frac{3}{4}\) and \(-\frac{3}{4}\), thus when we input these values into our polynomial, they would cause it to equal zero.

Factoring a polynomial to find its zeros is a critical skill, as it enables one to rewrite the polynomial in an understandable way that clearly shows its roots, helping with functions analysis and graphing.

Third Degree Polynomial

A third degree polynomial, also known as a cubic polynomial, is characterized by having the highest exponent of 'x' as three. This type of polynomial takes the general form \(ax^3+bx^2+cx+d\), where 'a', 'b', 'c', and 'd' are constants and \(a eq 0\).

The significance of a third degree polynomial lies in its potential to have up to three real zeros and it introduces more complexity in its graph, with the possibility of having multiple turning points. Take the original exercise's solution: \(p(x) = (x^2 - \frac{9}{16}) \cdot (x - \frac{3}{4})\). Here, we followed through step 3 to create a third degree polynomial, which presents a zero at \(\pm \frac{3}{4}\) and includes a repeated zero which affects the graph's nature at that point.

Fourth Degree Polynomial

A polynomial of fourth degree, often called a quartic polynomial, has its highest variable 'x' raised to the power of four. It can be represented as \(ax^4+bx^3+cx^2+dx+e\), with constants 'a', 'b', 'c', 'd', and 'e', and with \(a eq 0\).

Quartic polynomials are more complex than third degree ones as they can have up to four real zeros and may have more intricate curves and turning points in their graph. In our provided solution's context, \(p(x) = (x^2 - \frac{9}{16})^2\) is the fourth degree polynomial that satisfies the condition of having zeros at \(\pm \frac{3}{4}\), both repeated twice. This repetition implies the graph of the polynomial will touch the x-axis at these points, turning around rather than crossing it.