Simplify 2h+3h^2-4g^4+5f^2+11g^4-4h+f^2
The given problem is an algebraic expression that requires simplification. The task involves combining like terms, which are terms that have the same variables raised to the same power. The variables in the expression are h, g, and f, and they are involved in different powers. The goal is to add or subtract the coefficients (numerical factors) of these like terms to simplify the expression down to the smallest number of terms possible. Each different variable and its associated exponents form a unique type of term that must be combined separately with any matching types.
$2 h + 3 h^{2} - 4 g^{4} + 5 f^{2} + 11 g^{4} - 4 h + f^{2}$
Answer
Solution:
Simplification Process
Step 1:
Combine like terms by subtracting $4h$ from $2h$ to get $-2h$. The new expression is $3h^2 - 4g^4 + 5f^2 + 11g^4 - 2h + f^2$.
Step 2:
Combine the terms involving $g^4$ by adding $-4g^4$ to $11g^4$, resulting in $7g^4$. The expression now reads $3h^2 + 7g^4 + 5f^2 - 2h + f^2$.
Step 3:
Combine the terms involving $f^2$ by adding $5f^2$ to $f^2$, which gives $6f^2$. The expression simplifies to $3h^2 + 7g^4 + 6f^2 - 2h$.
Step 4:
Rearrange the terms to place $3h^2$ at the end, yielding the simplified expression $7g^4 + 6f^2 + 3h^2 - 2h$.
Knowledge Notes:
To simplify an algebraic expression, you follow these steps:
Combine like terms: Like terms are terms that contain the same variables raised to the same power. In the given expression, like terms are combined by addition or subtraction as appropriate.
Rearrange terms: Sometimes, it is customary to rearrange the terms in descending order of their degree or alphabetically. This step is more about the presentation than changing the mathematical value of the expression.
Use of algebraic identities: In some cases, simplification may involve using algebraic identities, though it is not necessary in this particular problem.
Maintain the equivalence: Throughout the simplification process, it is crucial to maintain the equivalence of the expression by performing the same operation on both sides of an equation if applicable.
Apply the distributive property: If necessary, distribute multiplication over addition or subtraction to simplify further.
Check your work: After simplifying, it's a good practice to review each step to ensure no errors were made in the process.
In this problem, the simplification process primarily involved combining like terms and rearranging them for clarity. No complex algebraic identities or distributive properties were needed.
