Question

factor to find all x-intercepts of the function.\\(f(x) = 3x^6 - 5x^5 - 48x^4 + 80x^3\\)

Explanation:

Step1: Factor out the greatest common factor (GCF)

The GCF of the terms \(3x^{6}\), \(-5x^{5}\), \(-48x^{4}\), and \(80x^{3}\) is \(x^{3}\). Factoring out \(x^{3}\) from each term:
\(f(x)=x^{3}(3x^{3}-5x^{2}-48x + 80)\)

Step2: Group the terms in the cubic polynomial

Group the first two terms and the last two terms in \(3x^{3}-5x^{2}-48x + 80\):
\(3x^{3}-5x^{2}-48x + 80=(3x^{3}-5x^{2})+(-48x + 80)\)

Step3: Factor out the GCF from each group

From the first group \(3x^{3}-5x^{2}\), the GCF is \(x^{2}\), so \(3x^{3}-5x^{2}=x^{2}(3x - 5)\).
From the second group \(-48x + 80\), the GCF is \(-16\), so \(-48x + 80=-16(3x - 5)\).
Now the expression becomes:
\((3x^{3}-5x^{2})+(-48x + 80)=x^{2}(3x - 5)-16(3x - 5)\)

Step4: Factor out the common binomial factor

We can factor out \((3x - 5)\) from \(x^{2}(3x - 5)-16(3x - 5)\):
\(x^{2}(3x - 5)-16(3x - 5)=(3x - 5)(x^{2}-16)\)

Step5: Factor the difference of squares

The term \(x^{2}-16\) is a difference of squares, which factors as \((x - 4)(x + 4)\) (since \(a^{2}-b^{2}=(a - b)(a + b)\) with \(a = x\) and \(b = 4\)).
So now we have:
\(3x^{3}-5x^{2}-48x + 80=(3x - 5)(x - 4)(x + 4)\)

Step6: Combine all the factors

Putting it all together with the \(x^{3}\) we factored out earlier:
\(f(x)=x^{3}(3x - 5)(x - 4)(x + 4)\)

Step7: Find the x - intercepts

To find the x - intercepts, we set \(f(x)=0\). So:
\(x^{3}(3x - 5)(x - 4)(x + 4)=0\)
Using the zero - product property (if \(ab = 0\), then either \(a = 0\) or \(b = 0\)), we get:

• For \(x^{3}=0\), we have \(x = 0\) (with multiplicity 3).
• For \(3x - 5=0\), solving for \(x\) gives \(x=\frac{5}{3}\).
• For \(x - 4=0\), we have \(x = 4\).
• For \(x + 4=0\), we have \(x=-4\).

Answer:

The x - intercepts of the function \(f(x)\) are \(x = 0\), \(x=-4\), \(x = 4\), and \(x=\frac{5}{3}\) (with \(x = 0\) having a multiplicity of 3). In terms of the solutions to \(f(x)=0\), the x - intercepts are \(\boldsymbol{-4}\), \(\boldsymbol{0}\), \(\boldsymbol{\frac{5}{3}}\), and \(\boldsymbol{4}\).