(x%5E4%2B3x%5E2%2B4x).jpeg "(4x^2-8x-2)(x^4+3x^2+4x)")
Multiplying Polynomials: (4x^2 - 8x - 2)(x^4 + 3x^2 + 4x)
This problem involves multiplying two polynomials. There are a couple of different ways to approach this, but the most common and reliable is using the distributive property.
Understanding the Distributive Property
The distributive property states that for any numbers a, b, and c:
a(b + c) = ab + ac*
This means we can distribute a term across a sum by multiplying it by each term inside the parentheses.
Applying the Distributive Property
To multiply (4x^2 - 8x - 2)(x^4 + 3x^2 + 4x), we can think of it as distributing each term of the first polynomial across the second polynomial:
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Distribute 4x^2: (4x^2)(x^4 + 3x^2 + 4x) = 4x^6 + 12x^4 + 16x^3
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Distribute -8x: (-8x)(x^4 + 3x^2 + 4x) = -8x^5 - 24x^3 - 32x^2
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Distribute -2: (-2)(x^4 + 3x^2 + 4x) = -2x^4 - 6x^2 - 8x
Combining Like Terms
Now, we have a sum of multiple terms. We need to combine the terms with the same powers of x:
4x^6 + 12x^4 + 16x^3 - 8x^5 - 24x^3 - 32x^2 - 2x^4 - 6x^2 - 8x
Combining like terms, we get:
4x^6 - 8x^5 + 10x^4 - 8x^3 - 38x^2 - 8x
Conclusion
The product of (4x^2 - 8x - 2)(x^4 + 3x^2 + 4x) is 4x^6 - 8x^5 + 10x^4 - 8x^3 - 38x^2 - 8x.
Remember, always combine like terms to simplify your polynomial expressions.