(-4x%5E2-2x-1).jpeg "(3x^2+x-2)(-4x^2-2x-1)")
Multiplying Polynomials: A Step-by-Step Guide
This article will explore the multiplication of two trinomials: (3x² + x - 2) and (-4x² - 2x - 1). We will use the distributive property and a systematic approach to arrive at the final product.
Understanding the Distributive Property
The distributive property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. We can apply this property to multiply polynomials by distributing each term of one polynomial across all the terms of the other.
Multiplying the Trinomials
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Expansion: First, we expand the first trinomial (3x² + x - 2) by multiplying each of its terms by the second trinomial (-4x² - 2x - 1).
(3x² + x - 2) * (-4x² - 2x - 1) = 3x² * (-4x² - 2x - 1) + x * (-4x² - 2x - 1) - 2 * (-4x² - 2x - 1)
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Distributing: Next, we distribute each term within the brackets.
= -12x⁴ - 6x³ - 3x² - 4x³ - 2x² - x + 8x² + 4x + 2
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Combining Like Terms: Finally, we combine like terms to simplify the expression.
= -12x⁴ - 10x³ + 3x² + 3x + 2
The Final Product
Therefore, the product of (3x² + x - 2) and (-4x² - 2x - 1) is -12x⁴ - 10x³ + 3x² + 3x + 2.
Conclusion
By following the steps outlined above, we successfully multiplied the two trinomials. The distributive property and a methodical approach are key to achieving accurate results. This process is applicable to multiplying any polynomials, allowing us to efficiently find the product of even complex expressions.