Question

question
factor to find all x-intercepts of the function,
$f(x) = x^4 + 8x$
answer attempt 1 out of a
additional solution no solution

Explanation:

Step1: Factor out the GCF

The greatest common factor (GCF) of \(x^4\) and \(8x\) is \(x\). So we factor out \(x\) from the function:
\(f(x)=x(x^3 + 8)\)

Step2: Factor the sum of cubes

Recall the sum of cubes formula \(a^3 + b^3=(a + b)(a^2 - ab + b^2)\). Here, \(a = x\) and \(b = 2\) (since \(2^3=8\)), so \(x^3 + 8=(x + 2)(x^2 - 2x + 4)\). Now our function is:
\(f(x)=x(x + 2)(x^2 - 2x + 4)\)

Step3: Find x - intercepts by setting \(f(x)=0\)

Set each factor equal to zero:

• For \(x = 0\), we have a solution \(x = 0\).
• For \(x+2=0\), we get \(x=-2\).
• For \(x^2 - 2x + 4=0\), we use the quadratic formula \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b=-2\), and \(c = 4\). The discriminant \(\Delta=b^2-4ac=(-2)^2-4\times1\times4=4 - 16=-12<0\), so this quadratic has no real solutions.

Answer:

The x - intercepts are \(x = 0\) and \(x=-2\)