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)=x3+7x2−4x−28f(x)=$$x^3$+7x^2$$−4x−28

Accepted Answer

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 $$(x - 	ext{zero}) to $$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.

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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Key Concepts

Finding Zeros of a Polynomial

Multiplicity of Zeros

Behavior of the Graph at Zeros

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