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)=3(x+5)(x+2)²

Verified step by step guidance

1. Identify the zeros of the polynomial function by setting each factor equal to zero. For the function \(f(x) = 3(x+5)(x+2)^2\), set \(x+5=0\) and \(x+2=0\) to find the zeros.
2. Solve each equation for \(x\): from \(x+5=0\), we get \(x=-5\); from \(x+2=0\), we get \(x=-2\). These are the zeros of the function.
3. Determine the multiplicity of each zero by looking at the exponent of the corresponding factor. The factor \((x+5)\) has an exponent of 1, so the zero \(x=-5\) has multiplicity 1. The factor \((x+2)^2\) has an exponent of 2, so the zero \(x=-2\) has multiplicity 2.
4. Interpret the multiplicity to understand the graph behavior at each zero: if the multiplicity is odd (like 1), the graph crosses the x-axis at that zero; if the multiplicity is even (like 2), the graph touches the x-axis and turns around at that zero.
5. Summarize the results: zero \(x=-5\) with multiplicity 1 means the graph crosses the x-axis at \(x=-5\), and zero \(x=-2\) with multiplicity 2 means the graph touches the x-axis and turns around at \(x=-2\).

Key Concepts

Zeros of a Polynomial Function

Zeros of a polynomial function are the values of x that make the function equal to zero. They correspond to the x-intercepts of the graph. To find zeros, set the function equal to zero and solve for x, often by factoring or using other algebraic methods.

Multiplicity of Zeros

Multiplicity refers to the number of times a particular zero appears as a factor in the polynomial. For example, if (x + 2)^2 is a factor, then x = -2 is a zero with multiplicity 2. Multiplicity affects the behavior of the graph at that zero.

Graph Behavior at Zeros Based on Multiplicity

The graph crosses the x-axis at zeros with odd multiplicity and touches the x-axis and turns around at zeros with even multiplicity. This means the shape of the graph near each zero depends on whether the multiplicity is odd or even.