To factorize the expression \( 2x^3 + 13x^2 + 15x \), follow these steps:

1. **Factor out the greatest common factor (GCF):**

   The GCF of all the terms is \( x \), so we can factor out \( x \):
   \[
   2x^3 + 13x^2 + 15x = x(2x^2 + 13x + 15)
   \]

2. **Factor the quadratic \( 2x^2 + 13x + 15 \):**

   We need to find two numbers that multiply to \( 2 \times 15 = 30 \) and add to \( 13 \).

   These numbers are \( 10 \) and \( 3 \), since \( 10 \times 3 = 30 \) and \( 10 + 3 = 13 \).

3. **Rewrite the middle term:**

   Now rewrite the middle term \( 13x \) as \( 10x + 3x \):
   \[
   2x^2 + 13x + 15 = 2x^2 + 10x + 3x + 15
   \]

4. **Group the terms and factor:**

   Group the first two terms and the last two terms:
   \[
   (2x^2 + 10x) + (3x + 15)
   \]

   Factor out the common factor from each group:
   \[
   2x(x + 5) + 3(x + 5)
   \]

5. **Factor out the common binomial:**

   Now factor out the common binomial \( (x + 5) \):
   \[
   (2x + 3)(x + 5)
   \]

6. **Final factorization:**

   Thus, the fully factored form of \( 2x^3 + 13x^2 + 15x \) is:
   \[
   x(2x + 3)(x + 5)
   \]

Let me know if you'd like more details on any part of the process!

---

Here are 5 related questions for further exploration:
1. How would you factor \( 3x^3 + 18x^2 + 12x \)?
2. What is the factorization of \( 4x^3 + 20x^2 + 25x \)?
3. Can you apply the same method to factor \( x^3 + 6x^2 + 9x \)?
4. How do you factor a quadratic expression like \( 6x^2 + 19x + 10 \)?
5. What is the next step if the quadratic can't be factored easily?

**Tip:** Always look for the greatest common factor (GCF) first when factoring any polynomial.

Learn how to factor the polynomial 2x³ + 13x² + 15x using the greatest common factor (GCF) and factoring quadratic expressions. This step-by-step guide covers algebraic concepts suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation