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)=2(x−5)(x+4)²

Verified step by step guidance

1. Identify the zeros of the polynomial by setting each factor equal to zero: solve \$2(x-5)(x+4)^2 = 0\( by setting \)x-5=0\( and \)x+4=0$.
2. From \(x-5=0\), find the zero \(x=5\). From \(x+4=0\), find the zero \(x=-4\).
3. Determine the multiplicity of each zero by looking at the exponents of the factors: the factor \((x-5)\) has an exponent of 1, so the zero \(x=5\) has multiplicity 1; the factor \((x+4)^2\) has an exponent of 2, so the zero \(x=-4\) has multiplicity 2.
4. Interpret the multiplicity to describe 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 at \(x=5\) with multiplicity 1 (graph crosses the x-axis), and zero at \(x=-4\) with multiplicity 2 (graph touches and turns around at the x-axis).

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. Finding zeros involves setting the function equal to zero and solving 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. A zero with multiplicity 1 is called a simple zero, while higher multiplicities indicate repeated roots. The 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, influencing how the function behaves visually.