Question: Expand the product $ (2x - 3)(x + 4)(x - 1) $. - Deep Underground Poetry
Expanding the Product: $ (2x - 3)(x + 4)(x - 1) $
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
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Key Insights
Step 1: Multiply the first two binomials
Start by multiplying $ (2x - 3) $ and $ (x + 4) $:
$$
(2x - 3)(x + 4) = 2x(x) + 2x(4) - 3(x) - 3(4)
$$
$$
= 2x^2 + 8x - 3x - 12
$$
$$
= 2x^2 + 5x - 12
$$
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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 $
