Factor the polynomial completely.
x Superscript 4 Baseline minus 55 x squared minus 6 x cubed


To factor the polynomial \(x^4 - 55x^2 - 6x^3\) completely, let's go through the steps systematically.

### Step 1: Arrange the Terms
First, rearrange the polynomial in standard form:
\[
x^4 - 6x^3 - 55x^2
\]

### Step 2: Factor Out the Greatest Common Factor (GCF)
The first step in factoring is to look for a common factor in all the terms. In this case, each term has a common factor of \(x^2\):
\[
x^2(x^2 - 6x - 55)
\]

### Step 3: Factor the Quadratic Trinomial
Now, focus on factoring the quadratic trinomial \(x^2 - 6x - 55\). We need to find two numbers that multiply to \(-55\) (the constant term) and add up to \(-6\) (the coefficient of the linear term).

Let's check the factors of \(-55\):
\[
-55 = (-11) \times 5
\]

Here, \(-11 + 5 = -6\), which matches the middle term.

So, we can factor the quadratic as:
\[
x^2 - 6x - 55 = (x - 11)(x + 5)
\]

### Step 4: Combine the Factors
Now, substitute the factored form of the quadratic back into the original expression:
\[
x^2(x - 11)(x + 5)
\]

### Final Answer
The completely factored form of the polynomial \(x^4 - 6x^3 - 55x^2\) is:
\[
x^2(x - 11)(x + 5)
\]

Would you like any more details or have any questions?

Here are 5 related questions you might find useful:

1. How would you factor a polynomial where the quadratic trinomial doesn't factor easily?
2. What is the process for factoring higher-degree polynomials?
3. Can the method of factoring by grouping be applied to this polynomial?
4. How do you determine if a polynomial can be factored further?
5. How can you use synthetic division to assist in factoring polynomials?

**Tip:** Always factor out the greatest common factor (GCF) first when factoring polynomials to simplify the process.

Learn how to factor the polynomial x^4 - 6x^3 - 55x^2 completely using step-by-step instructions. This problem involves factoring higher-degree polynomials and quadratic equations. Suitable for students in Grades 9-12.

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