Problem 52 - 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. \(f(x)=x^{2}+10 x+25\)

Step-by-Step Solution

Answer

The real zero of the function \(f(x)=x^{2}+10x+25\) is -5 and its multiplicity is 2.

Step 1: Set the function equal to zero

To find the zeros, we first need to set the given function equal to zero: \(x^{2}+10x+25=0\)

Step 2: Solve the equation

Next, we solve the equation. As this is a perfect square trinomial, we can write it as \((x+5)^{2}=0\). Thus, solving for 'x' gives us \(x+5=0\) or \(x=-5\)

Step 3: Determine the multiplicity of each zero

Since \(x=-5\) is obtained from the factor \((x+5)^{2}\), we say that the zero -5 has a multiplicity of 2. This means it appears twice as a factor in the factorized form of the polynomial.

Step 4: Verify with a graphing utility

We can check our answer using a graphing utility. When we plot the function \(f(x)=x^{2}+10x+25\), we should observe that the graph of the function intersects the x-axis at \(x=-5\) and the shape of the curve should give an indication of even multiplicity (the curve should touch and immediately turn around at the x-intercept, rather than cross through it).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special kind of quadratic polynomial that can be expressed in the form of a squared binomial. For a trinomial to be a perfect square, it follows the pattern \[ a^2 + 2ab + b^2 = (a + b)^2 \] This trinomial can be factored into the square of a binomial, allowing the polynomial to be neatly simplified.

• In our example, the polynomial is \(x^2 + 10x + 25\).
• It compares to the form where \(a = x\) and \(b = 5\).
• The middle term confirms it as it is twice the product of \(a\) and \(b\) (i.e., \(2 \cdot x \cdot 5 = 10x\)).

This property makes solving for zeros straightforward. You identify it as a squared term, leading you directly to \[ (x + 5)^2 = 0 \] which is easier to handle for solving the equation. Understanding this concept simplifies many quadratic problems and is a powerful tool for algebra students.

Zero Multiplicity

The concept of zero multiplicity refers to how many times a particular solution appears as a zero of the polynomial. In simple terms, if you rewrite the polynomial as a product of its factors, the multiplicity is the number of times a zero is repeated.

• In our function \(f(x) = (x+5)^2 = 0\), the zero is \(x = -5\).
• Since \(x + 5\) appears squared, \(x = -5\) is repeated twice as a solution.
• Thus, \(x = -5\) has a multiplicity of 2.

The significance of knowing the multiplicity lies in how it affects the graph of a polynomial:

• A zero with odd multiplicity indicates that the graph crosses the x-axis at that point.
• A zero with even multiplicity means the graph merely touches the x-axis at that zero and turns back, indicating a local minimum or maximum.

Understanding multiplicity gives insight into the behavior of polynomial functions at different intercepts.

Graphing Polynomial Functions

Graphing polynomial functions helps visualize their behavior and verify algebraic solutions. It is especially useful for confirming the zeros and understanding how the polynomial behaves around those zeros. Let's take a closer look.

• For our example, \(f(x) = x^2 + 10x + 25\), we turned it into \((x+5)^2\) which indicates a perfect square.
• When plotted, the graph will be a parabola opening upwards, which touches the x-axis at \(x = -5\).
• This touch point reflects the zero \[ x = -5 \] with an even multiplicity of 2, meaning the curve does not cross the axis but rather touches and turns around.

Graphing utilities can be employed to sketch these curves accurately and validate the algebraic findings. Key to understanding polynomial graphs is recognizing how zeros with varying multiplicities affect the curve's approach and interception of the x-axis. Visualizing this interaction enables a deeper comprehension of algebraic solutions and the dynamic nature of polynomial equations.