which statement describes the graph of this polynomial function?\n$f(x)=x^5 - 6x^4 + 9x^3$\n- the graph…

which statement describes the graph of this polynomial function?\n$f(x)=x^5 - 6x^4 + 9x^3$\n- the graph crosses the x-axis at $x = 0$ and touches the x-axis at $x = 3$.\n- the graph touches the x-axis at $x = 0$ and crosses the x-axis at $x = 3$.\n- the graph crosses the x-axis at $x = 0$ and touches the x-axis at $x = -3$.\n- the graph touches the x-axis at $x = 0$ and crosses the x-axis at $x = -3$.

which statement describes the graph of this polynomial function?\n$f(x)=x^5 - 6x^4 + 9x^3$\n- the graph crosses the x-axis at $x = 0$ and touches the x-axis at $x = 3$.\n- the graph touches the x-axis at $x = 0$ and crosses the x-axis at $x = 3$.\n- the graph crosses the x-axis at $x = 0$ and touches the x-axis at $x = -3$.\n- the graph touches the x-axis at $x = 0$ and crosses the x-axis at $x = -3$.

Answer

Explanation:

Step1: Factor the polynomial

First, factor out the greatest common factor (x^3) from (f(x) = x^5 - 6x^4 + 9x^3). So we get (f(x)=x^3(x^2 - 6x + 9)). Then, factor the quadratic expression (x^2 - 6x + 9), which is a perfect square trinomial and factors to ((x - 3)^2). So the factored form of the polynomial is (f(x)=x^3(x - 3)^2).

Step2: Analyze the roots and their multiplicities

For a polynomial (f(x)=(x - a)^n), if (n) is odd, the graph crosses the (x)-axis at (x = a); if (n) is even, the graph touches the (x)-axis at (x = a) (because the function doesn't change sign at that point).

  • For the root (x = 0), the multiplicity is (3) (since the factor is (x^3=(x - 0)^3)), and (3) is odd. So the graph crosses the (x)-axis at (x = 0).
  • For the root (x = 3), the multiplicity is (2) (since the factor is ((x - 3)^2)), and (2) is even. So the graph touches the (x)-axis at (x = 3).

Answer:

The graph crosses the x - axis at (x = 0) and touches the x - axis at (x = 3). (The first option: "The graph crosses the x - axis at (x = 0) and touches the x - axis at (x = 3)")