Factor completely. 4, x, cubed, minus, 28, x, squared, minus, 9, x, plus, 63 4x 3 −28x 2 −9x+63



The given expression is:

\[
4x^3 - 28x^2 - 9x + 63
\]

We can factor this step-by-step.

### Step 1: Group terms
Group the terms in pairs:

\[
(4x^3 - 28x^2) - (9x - 63)
\]

### Step 2: Factor out the greatest common factor (GCF) from each group
From the first group, \(4x^3 - 28x^2\), the GCF is \(4x^2\).
From the second group, \(-9x + 63\), the GCF is \(-9\).

Factoring these out gives:

\[
4x^2(x - 7) - 9(x - 7)
\]

### Step 3: Factor out the common binomial factor
Both terms have a common factor of \((x - 7)\), so we can factor it out:

\[
(x - 7)(4x^2 - 9)
\]

### Step 4: Factor the difference of squares
The term \(4x^2 - 9\) is a difference of squares, as it can be written as:

\[
4x^2 - 9 = (2x - 3)(2x + 3)
\]

### Final factored form:
\[
(x - 7)(2x - 3)(2x + 3)
\]

Thus, the complete factorization of the expression is:

\[
(x - 7)(2x - 3)(2x + 3)
\]

Let me know if you'd like any further explanation!

### Related Questions:
1. How do you recognize a difference of squares?
2. What is the general form of factoring by grouping?
3. How do you factor cubic polynomials using synthetic division?
4. How would the factorization change if there was a different constant term?
5. Can every cubic polynomial be factored by grouping?

### Tip:
Always check for a GCF (greatest common factor) first, as this simplifies the factoring process significantly.

Factor completely. 4x^3 - 28x^2 - 9x + 63

Learn how to factor the cubic polynomial 4x^3 - 28x^2 - 9x + 63 step-by-step. This solution covers key algebraic concepts, including factoring by grouping and the difference of squares. Perfect for students in Grades 9-11.

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