3Z^2 + 5 * 6Z^5 + 15Z^4 + 16Z^3 + 4Z^2 + 10Z – 35
The Correct Answer and Explanation is:
Let’s simplify and factor the expression:
Given expression:3Z2+5⋅6Z5+15Z4+16Z3+4Z2+10Z−353Z^2 + 5 \cdot 6Z^5 + 15Z^4 + 16Z^3 + 4Z^2 + 10Z – 353Z2+5⋅6Z5+15Z4+16Z3+4Z2+10Z−35
Step 1: Simplify the expression
Multiply the constants:5⋅6Z5=30Z55 \cdot 6Z^5 = 30Z^55⋅6Z5=30Z5
Now rewrite the full expression in descending order of powers:30Z5+15Z4+16Z3+4Z2+10Z−35+3Z230Z^5 + 15Z^4 + 16Z^3 + 4Z^2 + 10Z – 35 + 3Z^230Z5+15Z4+16Z3+4Z2+10Z−35+3Z2
Combine like terms:30Z5+15Z4+16Z3+(4Z2+3Z2)+10Z−3530Z^5 + 15Z^4 + 16Z^3 + (4Z^2 + 3Z^2) + 10Z – 3530Z5+15Z4+16Z3+(4Z2+3Z2)+10Z−3530Z5+15Z4+16Z3+7Z2+10Z−3530Z^5 + 15Z^4 + 16Z^3 + 7Z^2 + 10Z – 3530Z5+15Z4+16Z3+7Z2+10Z−35
Final simplified polynomial:
30Z5+15Z4+16Z3+7Z2+10Z−3530Z^5 + 15Z^4 + 16Z^3 + 7Z^2 + 10Z – 3530Z5+15Z4+16Z3+7Z2+10Z−35
Explanation
To simplify and factor polynomials, the expression should first be organized in descending order of degree, with all like terms combined. This ensures clarity and correctness in further algebraic manipulation. In the given expression, the term 5 * 6Z^5 is a product that must be evaluated before combining with the rest of the polynomial. Multiplying gives 30Z^5.
Once simplified, the polynomial becomes:30Z5+15Z4+16Z3+7Z2+10Z−3530Z^5 + 15Z^4 + 16Z^3 + 7Z^2 + 10Z – 3530Z5+15Z4+16Z3+7Z2+10Z−35
This is a 5th-degree polynomial with six terms. Such expressions are called polynomials of higher degree and can sometimes be factored by grouping or using polynomial division. A good first step is to look for a greatest common factor (GCF). Here, no GCF exists across all terms, so factoring by grouping is attempted.
Group terms strategically:
- Group 1: (30Z5+15Z4)(30Z^5 + 15Z^4)(30Z5+15Z4)
- Group 2: (16Z3+7Z2)(16Z^3 + 7Z^2)(16Z3+7Z2)
- Group 3: (10Z−35)(10Z – 35)(10Z−35)
Factoring each group:
- 30Z5+15Z4=15Z4(2Z+1)30Z^5 + 15Z^4 = 15Z^4(2Z + 1)30Z5+15Z4=15Z4(2Z+1)
- 16Z3+7Z2=Z2(16Z+7)16Z^3 + 7Z^2 = Z^2(16Z + 7)16Z3+7Z2=Z2(16Z+7)
- 10Z−35=5(2Z−7)10Z – 35 = 5(2Z – 7)10Z−35=5(2Z−7)
Since the factored parts are not the same, factoring further is not straightforward. This suggests the polynomial may not factor nicely over integers and could require more advanced methods like rational root theorem, synthetic division, or numerical methods.
In conclusion, the expression simplifies to:30Z5+15Z4+16Z3+7Z2+10Z−35\boxed{30Z^5 + 15Z^4 + 16Z^3 + 7Z^2 + 10Z – 35}30Z5+15Z4+16Z3+7Z2+10Z−35
It does not factor easily, and unless specific instructions suggest further factoring, the simplified form is the final result.
