Problem 27 - Algebra and Trigonometry

Question

Find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the x@axis, or touches the x@axis and turns around, at each zero. \(f(x)=4(x-3)(x+6)^{3}\)

Step-by-Step Solution

Answer

The zeros of the function \(f(x)=4(x-3)(x+6)^{3}\) are \(x=3\) with multiplicity 1 and \(x=-6\) with multiplicity 3. The graph crosses the x-axis at both zeros.

Step 1: Find the Zeros

Set the function equal to zero: \(4(x-3)(x+6)^{3} = 0\). Solve for \(x\) by isolating each factor. The equation given is already factored, so we can set each factor equal to zero: \(x-3=0\) and \(x+6=0\). Solving these results in \(x=3\) and \(x=-6\). These two values are the zeros of \(f(x)\).

Step 2: Determine the Multiplicity

The multiplicity of a zero is determined by the exponent with which it appears in the factored form of the polynomial. Here, \(3\) appears once (since its factor is taken to the power of 1) and \(-6\) appears three times (since the factor is taken to the power of 3). Therefore, the multiplicity of \(3\) is 1 and the multiplicity of \(-6\) is 3.

Step 3: Determine behaviour at the Zeros

When the graph reaches a zero, its behaviour depends on the multiplicity of that zero. If the multiplicity is odd, the graph crosses the x-axis. If the multiplicity is even, the graph touches the x-axis and turns around. The multiplicity of \(3\) (which is 1 and thus odd) means the graph crosses the x-axis at \(x=3\). However, since \(-6\) has a multiplicity of 3 (which is also odd), the graph also crosses the x-axis at \(x=-6\).

Key Concepts

Zeros of a Polynomial

The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. Finding these zeros is crucial since they provide critical information about the graph of the polynomial on a coordinate plane.
For the polynomial function given as \( f(x) = 4(x - 3)(x + 6)^3 \), setting \( f(x) = 0 \) helps identify the zeros. Each factor of the polynomial equation indicates a potential zero.

• For the factor \( (x - 3) \), setting \( x - 3 = 0 \) gives us the zero \( x = 3 \).
• For the factor \( (x + 6)^3 \), setting \( x + 6 = 0 \) results in the zero \( x = -6 \).

Thus, the zeros of the polynomial function are \( x = 3 \) and \( x = -6 \), signifying points where the graph of the polynomial will intersect the x-axis.

Multiplicity of a Zero

Multiplicity refers to the number of times a particular zero is a solution of the polynomial equation. It offers insights into the touching or crossing action at a zero.
The polynomial \( f(x) = 4(x - 3)(x + 6)^3 \) has zeros with specific multiplicities:

• The zero \( x = 3 \) arises from the factor \( (x - 3) \), which appears only once, indicating a multiplicity of 1.
• The zero \( x = -6 \) is derived from the factor \( (x + 6) \), raised to the power of 3, denoting a multiplicity of 3.

Understanding multiplicity is vital for graphing as it determines whether the graph will just touch the axis at the zero or cross it.

Graph Behavior at Zeros

The behavior of the graph at zeros of a polynomial is influenced by the multiplicity of those zeros. Here's what to expect:

• If the multiplicity of the zero is odd, the graph will cross the x-axis at that zero.
• If the multiplicity is even, the graph will touch the x-axis and turn around without crossing it.

For \( f(x) = 4(x - 3)(x + 6)^3 \):

• The zero \( x = 3 \) has an odd multiplicity of 1; thus, the graph crosses the x-axis at this point.
• The zero \( x = -6 \) has an odd multiplicity of 3, indicating the graph also crosses the x-axis at this zero.

Knowing how a polynomial function behaves at its zeros can assist in sketching an accurate graph.