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Answer :
Final answer:
The completely factored form of the polynomial 2x⁶ – 7x⁵ – 15x⁴ is x⁴(x – 10)(2x + 3).
Explanation:
To factor the polynomial 2x⁶ – 7x⁵ – 15x⁴ completely, we first look for the greatest common factor (GCF) among the terms. In this case, the GCF is x⁴. We can factor out x⁴ from each term:
2x⁶ – 7x⁵ – 15x⁴ = x⁴(2x² – 7x – 15)
Now, we have a quadratic expression inside the parentheses: 2x² – 7x – 15. To factor this quadratic expression, we look for two numbers whose product is -30 (the product of the leading coefficient and the constant term) and whose sum is -7 (the coefficient of the middle term).
After some trial and error, we find that the numbers -10 and 3 satisfy these conditions. We can rewrite the quadratic expression as:
2x² – 7x – 15 = (x – 10)(2x + 3)
Therefore, the completely factored form of the polynomial 2x⁶ – 7x⁵ – 15x⁴ is:
2x⁶ – 7x⁵ – 15x⁴ = x⁴(x – 10)(2x + 3)
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