Final answer:
The polynomial x^4 + 7x^3 + 10x^2 is factored by first taking out the GCF, which is x^2, resulting in x^2(x^2 + 7x + 10). Then, the quadratic is factored further into (x + 5)(x + 2), giving the final factorized form as x^2(x + 5)(x + 2).
Step-by-step explanation:
To factor out the greatest common factor (GCF) from the polynomial x^4 + 7x^3 + 10x^2, first identify the GCF of the terms. The GCF here is x^2, as it is the highest power of x that divides all three terms.
Factoring out the GCF, we have:
x^2(x^2 + 7x + 10).
Next, we can complete the factorization by factoring the quadratic expression within the parentheses. The quadratic x^2 + 7x + 10 can be factored into:
x^2(x + 5)(x + 2)
This gives us the fully factored form of the polynomial. It is crucial to check if the factorization is reasonable.
By distributing the factors, one should be able to arrive back at the original polynomial, which is a good way to verify the factorization.