Expanding the Product: $ (2x - 3)(x + 4)(x - 1) $

If you're working with cubic expressions in algebra, expanding products like $ (2x - 3)(x + 4)(x - 1) $ may seem tricky at first—but with the right approach, it becomes a smooth process. In this article, we’ll walk step-by-step through expanding the expression $ (2x - 3)(x + 4)(x - 1) $, explain key algebraic concepts, and highlight how mastering this technique improves your overall math proficiency.

Understanding the Context

Why Expand Algebraic Expressions?

Expanding products helps simplify expressions, solve equations, and prepare for higher-level math such as calculus and polynomial factoring. Being able to expand $ (2x - 3)(x + 4)(x - 1) $ not only aids in solving expressions but also strengthens problem-solving skills.

Step-by-Step Expansion

Question: Expand the product $ (2x - 3)(x + 4)(x - 1) $.

Key Insights

Step 1: Multiply the first two binomials

Start by multiplying $ (2x - 3) $ and $ (x + 4) $:

$$
(2x - 3)(x + 4) = 2x(x) + 2x(4) - 3(x) - 3(4)
$$

$$
= 2x^2 + 8x - 3x - 12
$$

$$
= 2x^2 + 5x - 12
$$

Continue Reading

Use the identity:
We know \( (a + b)^3 = 12^3 = 1728 \). To use the identity, we need \( ab \). From the identity:
= 12^2 - 2ab = 144 - 2ab

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Final Thoughts

Step 2: Multiply the result by the third binomial

Now multiply $ (2x^2 + 5x - 12)(x - 1) $:

Use the distributive property (also known as FOIL for binomials extended to polynomials):

$$
(2x^2 + 5x - 12)(x - 1) = 2x^2(x) + 2x^2(-1) + 5x(x) + 5x(-1) -12(x) -12(-1)
$$

$$
= 2x^3 - 2x^2 + 5x^2 - 5x - 12x + 12
$$

Step 3: Combine like terms

Now combine terms with the same degree:

  • $ 2x^3 $
  • $ (-2x^2 + 5x^2) = 3x^2 $
  • $ (-5x - 12x) = -17x $
  • Constant: $ +12 $