Multiplying the Binomials (3x+4)(2x+3)(5x+4)(x+2): A Step‑by‑Step Guide
The moment you see the string “3x 4 2x 3 5x 4 x 2” in an algebra problem, it usually shorthand for the product of four binomials: (3x + 4)(2x + 3)(5x + 4)(x + 2). Understanding how to expand this expression builds a solid foundation for polynomial manipulation, which is essential in everything from solving quadratic equations to modeling real‑world phenomena. Below is a detailed, easy‑to‑follow walkthrough that breaks down each multiplication stage, highlights common pitfalls, and offers practice ideas to reinforce your skills Simple, but easy to overlook..
Introduction: Why This Expression Matters
The expression (3x + 4)(2x + 3)(5x + 4)(x + 2) appears frequently in textbook exercises that test the distributive property and the FOIL method. Mastering it helps you:
- Recognize patterns in polynomial multiplication.
- Develop fluency with combining like terms.
- Prepare for higher‑level topics such as factoring, polynomial division, and calculus.
By the end of this article, you’ll be able to expand the product confidently and verify your result using alternative methods Worth keeping that in mind..
Understanding the Components
Before jumping into multiplication, let’s clarify what each part represents.
| Binomial | Meaning |
|---|---|
| 3x + 4 | A linear term with coefficient 3 and constant 4. Consider this: |
| 2x + 3 | Another linear term; coefficient 2, constant 3. |
| 5x + 4 | Linear term with coefficient 5, constant 4. |
| x + 2 | Simplest binomial; coefficient 1 for x, constant 2. |
This changes depending on context. Keep that in mind.
Each binomial is of the form ax + b, where a and b are real numbers. Multiplying four of them yields a polynomial of degree 4 (since the highest power of x comes from multiplying the x‑terms together: x·x·x·x = x⁴).
Step‑by‑Step Multiplication
1. Multiply the First Two Binomials
We start with (3x + 4)(2x + 3). Using the distributive property (or FOIL):
- First: 3x · 2x = 6x²
- Outer: 3x · 3 = 9x
- Inner: 4 · 2x = 8x
- Last: 4 · 3 = 12
Combine the like terms (9x + 8x = 17x):
[ (3x+4)(2x+3) = 6x^{2} + 17x + 12 ]
2. Multiply the Next Pair
Now handle (5x + 4)(x + 2):
- First: 5x · x = 5x²
- Outer: 5x · 2 = 10x
- Inner: 4 · x = 4x
- Last: 4 · 2 = 8
Combine the x‑terms (10x + 4x = 14x):
[ (5x+4)(x+2) = 5x^{2} + 14x + 8 ]
3. Multiply the Two Quadratic Results
We now need to multiply the two trinomials obtained:
[ (6x^{2} + 17x + 12)(5x^{2} + 14x + 8) ]
Because each factor has three terms, we will distribute each term of the first polynomial across the second polynomial. It’s helpful to organize the work in a table or use vertical multiplication Worth knowing..
Distribution Breakdown
| Term from First Polynomial | × Each Term of Second Polynomial | Result |
|---|---|---|
| 6x² | 6x²·5x² = 30x⁴ <br> 6x²·14x = 84x³ <br> 6x²·8 = 48x² | 30x⁴ + 84x³ + 48x² |
| 17x | 17x·5x² = 85x³ <br> 17x·14x = 238x² <br> 17x·8 = 136x | 85x³ + 238x² + 136x |
| 12 | 12·5x² = 60x² <br> 12·14x = 168x <br> 12·8 = 96 | 60x² + 168x + 96 |
Adding All Contributions
Now sum the like‑degree terms:
- x⁴: 30x⁴
- x³: 84x³ + 85x³ = 169x³
- x²: 48x² + 238x² + 60x² = 346x²
- x¹: 136x + 168x = 304x
- Constant: 96
Thus the expanded form is:
[ 30x^{4} + 169x^{3} + 346x^{2} + 3
