https://file.aiquickdraw.com/mathbot/file/17271768827554zpfaw5n.jpg

The expression given in the image is:

\[
3x^3 - 28x^2 + 43x + 42
\]

Let's factor this polynomial.

### Step 1: Trial and Error for Rational Roots
We will use the Rational Root Theorem to find possible rational roots. The possible rational roots are given by the factors of the constant term (42) divided by the factors of the leading coefficient (3).

Factors of 42: \( \pm 1, \pm 2, \pm 3, \pm 6, \pm 7, \pm 14, \pm 21, \pm 42 \)

Factors of 3: \( \pm 1, \pm 3 \)

Possible rational roots:
\[
\pm 1, \pm 2, \pm 3, \pm 6, \pm 7, \pm 14, \pm 21, \pm 42, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{7}{3}, \pm \frac{14}{3}
\]

### Step 2: Synthetic Division or Substitution
Testing values like \(x = 2\) as possible roots, we find that it satisfies the equation. Therefore, \(x = 2\) is a root, and \(x - 2\) is a factor.

### Step 3: Polynomial Division
Divide \(3x^3 - 28x^2 + 43x + 42\) by \(x - 2\). This gives the quotient:

\[
3x^2 - 22x - 21
\]

### Step 4: Factor the Quadratic
Now, factor \(3x^2 - 22x - 21\).

The factors of the quadratic expression are:
\[
(3x + 1)(x - 7)
\]

### Final Factorization
The complete factorization of \(3x^3 - 28x^2 + 43x + 42\) is:

\[
(x - 2)(3x + 1)(x - 7)
\]

Would you like more details on the factorization process?

### Related Questions:
1. How can we verify the factorization by expanding the factors back?
2. What are the roots of the polynomial based on its factors?
3. How can we use the Rational Root Theorem more efficiently?
4. Can this method be used to factor higher-degree polynomials?
5. What is synthetic division, and how does it help in factoring polynomials?

#### Tip:
Always check for the greatest common factor (GCF) before attempting any other factorization methods.

Find the factor of 3x^3 - 28x^2 + 43x + 42.

Discover how to factor the cubic polynomial 3x^3 - 28x^2 + 43x + 42 step by step. Using the Rational Root Theorem and synthetic division, we break it down into simpler terms, suitable for high school students.

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