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. $f (x) = x^{3} + 7 x^{2} - 4 x - 28$

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First, write down the polynomial function: $f(x) = x^3 + 7x^2 - 4x - 28$.

Use the Rational Root Theorem to list possible rational zeros. These are factors of the constant term ($-28$) divided by factors of the leading coefficient (1), so possible zeros are $\pm1, \pm2, \pm4, \pm7, \pm14, \pm28$.

Test these possible zeros by substituting them into $f(x)$ or by using synthetic division to find which values make $f(x) = 0$. Each value that results in zero is a root (zero) of the polynomial.

Once a zero is found, use polynomial division (synthetic or long division) to divide $f(x)$ by the corresponding factor $(x - \text{zero})$ to reduce the polynomial to a quadratic or lower degree polynomial.

Repeat the process on the reduced polynomial to find all zeros. For each zero, determine its multiplicity by how many times the factor appears in the factorization. If the multiplicity is odd, the graph crosses the x-axis at that zero; if even, the graph touches the x-axis and turns around.

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Finding Zeros of a Polynomial

Zeros of a polynomial are the values of x for which the function equals zero. To find them, you can use factoring, synthetic division, or the Rational Root Theorem. Identifying zeros is essential because they represent the x-intercepts of the graph.

Multiplicity of Zeros

Multiplicity refers to how many times a particular zero appears as a root of the polynomial. If a zero has an even multiplicity, the graph touches the x-axis and turns around at that point; if odd, the graph crosses the x-axis.

Behavior of the Graph at Zeros

The graph's behavior at each zero depends on the zero's multiplicity. For odd multiplicities, the graph crosses the x-axis, while for even multiplicities, it only touches and reverses direction. This helps in sketching the polynomial's graph accurately.

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)=x3+7x2−4x−28f(x)=x^3+7x^2−4x−28