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Answer :
Final answer:
The factorization of the polynomial 10x⁵-20x³+25x² is achieved by finding the common factor among all terms. This results in the expression 10x²(x³-2x+5).
Explanation:
To factor the polynomial 10x⁵-20x³+25x², we first search for the common factor among all terms. In this case, the common factor is 5x².
By dividing all the terms by this common factor, we get: 5x²(2x³-4x+5).
However, we also see that the remaining terms within the parentheses have a common factor of 2. In this case, we can further factor out the 2 to get: 5x² x 2(x³-2x+5).
Thus, the complete factoring of the polynomial 10x⁵-20x³+25x² is 10x²(x³-2x+5).
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Final answer:
The polynomial cannot be factored further over the set of real numbers, as the trinomial expression 2x³ - 4x + 5 does not have real roots. The given factorization 5x²(2x³ - 4x + 5) is the most simplified form.
Explanation:
To factor the polynomial 10x⁵-20x³+25x² completely, we can start by factoring out the greatest common factor (GCF) of the terms, which in this case is 5x²:
10x⁵-20x³+25x² = 5x²(2x³ - 4x + 5).
Now, we need to factor the trinomial expression 2x³ - 4x + 5. To do this, we look for factors or roots of the expression. Unfortunately, in this case, the expression cannot be factored further using real numbers, and it does not have real roots.
Thus, the factorization is complete. The final factored form of the polynomial is 5x²(2x³ - 4x + 5).
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