To multiply the polynomials \((x^4 + 7)(5x^2 + 10x - 1)\), we will use the distributive property, often called the FOIL method for binomials, but it applies to polynomials of any degree as well. Here are the steps:
Steps to Multiply the Polynomials:
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Distribute each term in the first polynomial to each term in the second polynomial. This means you'll take each term in \(x^4 + 7\) and multiply it by each term in \(5x^2 + 10x - 1\).
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Multiply \(x^4\) by each term in the second polynomial:
- Multiply \(x^4 \cdot 5x^2 = 5x^{6}\)
- Multiply \(x^4 \cdot 10x = 10x^{5}\)
- Multiply \(x^4 \cdot (-1) = -x^{4}\)
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Multiply \(7\) by each term in the second polynomial:
- Multiply \(7 \cdot 5x^2 = 35x^{2}\)
- Multiply \(7 \cdot 10x = 70x\)
- Multiply \(7 \cdot (-1) = -7\)
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Combine all the results from the multiplications:
- From \(x^4\): \(5x^6, 10x^5, -x^4\)
- From \(7\): \(35x^2, 70x, -7\)
- So, we combine these: \[ 5x^6 + 10x^5 - x^4 + 35x^2 + 70x - 7 \]
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Write the final result in standard form (typically ordered by decreasing degree): \[ 5x^6 + 10x^5 - x^4 + 70x + 35x^2 - 7 \]
Final Answer:
The product of the polynomials \((x^4 + 7)(5x^2 + 10x - 1)\) is: \[ 5x^6 + 10x^5 - x^4 + 35x^2 + 70x - 7 \]
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