Answer :
To factor the expression [tex]\(2x^3 - 5x^2 + 25\)[/tex] completely, we need to consider possible factorization methods such as looking for common factors, factoring by grouping, or using any known formulas or theorems about factoring polynomials.
### Step 1: Check for a Common Factor
First, we should check if there is a common factor among all the terms in the polynomial. The polynomial [tex]\(2x^3 - 5x^2 + 25\)[/tex] has coefficients 2, -5, and 25. The greatest common factor of these numbers is 1, which means there is no common factor other than 1.
### Step 2: Try Factoring by Grouping
The expression is a cubic polynomial with three terms, which generally cannot be easily factored by grouping.
### Step 3: Use Factor Theorems or Trial and Error with Rational Root Theorem
For higher-degree polynomials, you might use the Rational Root Theorem to test potential rational roots. However, let's assume we've tested possible roots and found none work or result in integer factors.
### Step 4: Consider Completing the Factorization Over Real Numbers
Since we were unable to find roots that simplify the polynomial further, it is possible that the polynomial is already in its simplest form for factoring over the integers or rational numbers.
### Conclusion
In conclusion, the polynomial [tex]\(2x^3 - 5x^2 + 25\)[/tex] cannot be factored further using simple integer or rational factorization methods. Therefore, it remains [tex]\(2x^3 - 5x^2 + 25\)[/tex] as its completely factored form within the scope of the integers and rationals.
### Step 1: Check for a Common Factor
First, we should check if there is a common factor among all the terms in the polynomial. The polynomial [tex]\(2x^3 - 5x^2 + 25\)[/tex] has coefficients 2, -5, and 25. The greatest common factor of these numbers is 1, which means there is no common factor other than 1.
### Step 2: Try Factoring by Grouping
The expression is a cubic polynomial with three terms, which generally cannot be easily factored by grouping.
### Step 3: Use Factor Theorems or Trial and Error with Rational Root Theorem
For higher-degree polynomials, you might use the Rational Root Theorem to test potential rational roots. However, let's assume we've tested possible roots and found none work or result in integer factors.
### Step 4: Consider Completing the Factorization Over Real Numbers
Since we were unable to find roots that simplify the polynomial further, it is possible that the polynomial is already in its simplest form for factoring over the integers or rational numbers.
### Conclusion
In conclusion, the polynomial [tex]\(2x^3 - 5x^2 + 25\)[/tex] cannot be factored further using simple integer or rational factorization methods. Therefore, it remains [tex]\(2x^3 - 5x^2 + 25\)[/tex] as its completely factored form within the scope of the integers and rationals.
