which statement describes the graph of $f(x) = -4x^3 - 28x^2 - 32x + 64$?\nthe graph crosses the x-axis at…

which statement describes the graph of $f(x) = -4x^3 - 28x^2 - 32x + 64$?\nthe graph crosses the x-axis at $x = 4$ and touches the x-axis at $x = -1$.\nthe graph touches the x-axis at $x = 4$ and crosses the x-axis at $x = -1$.\nthe graph crosses the x-axis at $x = -4$ and touches the x-axis at $x = 1$.\nthe graph touches the x-axis at $x = -4$ and crosses the x-axis at $x = 1$.

which statement describes the graph of $f(x) = -4x^3 - 28x^2 - 32x + 64$?\nthe graph crosses the x-axis at $x = 4$ and touches the x-axis at $x = -1$.\nthe graph touches the x-axis at $x = 4$ and crosses the x-axis at $x = -1$.\nthe graph crosses the x-axis at $x = -4$ and touches the x-axis at $x = 1$.\nthe graph touches the x-axis at $x = -4$ and crosses the x-axis at $x = 1$.

Answer

Explanation:

Step1: Factor out the GCF

Factor $-4$ from the polynomial: $f(x) = -4(x^3 + 7x^2 + 8x - 16)$

Step2: Find rational roots

Use Rational Root Theorem to test $x=1$: $1^3 + 7(1)^2 + 8(1) - 16 = 1+7+8-16=0$ So $(x-1)$ is a factor.

Step3: Perform polynomial division

Divide $x^3 + 7x^2 + 8x -16$ by $(x-1)$: $x^3 + 7x^2 + 8x -16 = (x-1)(x^2 + 8x + 16)$

Step4: Factor quadratic

Factor the perfect square quadratic: $x^2 + 8x + 16 = (x+4)^2$

Step5: Write full factored form

$f(x) = -4(x-1)(x+4)^2$

Step6: Analyze x-intercept behavior

  • For root $x=1$ (multiplicity 1, odd): graph crosses x-axis.
  • For root $x=-4$ (multiplicity 2, even): graph touches x-axis.

Answer:

The graph touches the x-axis at $x = -4$ and crosses the x-axis at $x = 1$.