Problem 51 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all the real zeros of the polynomial function. Determine the multiplicity of each zero. Use a graphing utility to verify your results. \(h(t)=t^{2}-6 t+9\)

Step-by-Step Solution

Answer

The real zero of function \(h(t)=t^{2}-6t+9\) is \(t=3\) with a multiplicity of '2'

Step 1: Identify the Polynomial Function

The given polynomial function is \(h(t)=t^{2}-6 t+9\)

Step 2: Determine Real Zeros by Factoring

To find the real zeros, one would set the polynomial function equal to zero and solve for 't':\n\[0=t^{2}-6t+9\]\nThis can be factored as \((t-3)^2=0\) which gives a single solution of \(t = 3\)

Step 3: Determine the Multiplicity

The zero is '3' which has multiplicity '2'. This is because \((t-3)^2=0\) implies that '3' is a root of the equation two times.

Key Concepts

Real Zeros

Real zeros of a polynomial are the values of the variable that make the polynomial equal to zero. In the exercise given, you are asked to find the real zeros of the polynomial function \(h(t)=t^{2}-6t+9\). A real zero represents a point where the graph of the polynomial crosses or touches the x-axis.
To find these zeros, set the polynomial equation to zero and solve for the variable. Solving the equation \(0=t^{2}-6t+9\) involves factoring or using methods like the quadratic formula. When the polynomial is successfully factored, the solutions to the equation are the real zeros.
In this particular function, the factored form

• \((t-3)^2=0\) implies a real zero of \(t=3\).

Understanding this means recognizing how zeros relate to the graph: each zero gives a point where the polynomial can intersect the x-axis.

Multiplicity of Zeros

Multiplicity of a zero refers to the number of times a particular zero appears as a solution of the polynomial equation. When you factor the polynomial \((t-3)^2=0\), you find that the zero, \(t=3\), appears twice.
This is why we say that the zero has a multiplicity of 2. Multiplicity affects the shape of the graph of the polynomial at the zero.

• If a zero has an odd multiplicity, the graph crosses the x-axis at that zero.
• If it has an even multiplicity, the graph touches but does not cross the x-axis.

So for \(t=3\), the graph will touch the x-axis at this point, allowing us to visually verify the multiplicity when graphing.

Factoring Polynomials

Factoring polynomials is a key method used to find the real zeros of a polynomial function. It involves breaking down the polynomial into simpler "factor" polynomials that can be easily solved. For the exercise \(h(t)=t^{2}-6t+9\), the polynomial was factored into \((t-3)^2\). This means rewriting the original polynomial as a product of its factors.
This step is essential because it directly leads to finding the roots or zeros of the polynomial equation.

• Start by expressing the polynomial in its simplest form.
• Look for patterns or use methods like grouping to factor it.
• The factored form reveals the solutions or zeros easily.

With the factor \((t-3)^2\), you can quickly identify \(t=3\) as the zero, simplifying the process of solving the equation and understanding its graph.