Simplify (-5a^4-7a^6+3)+(3a^6+9-9a^4+5a^2)
The given problem is an expression that needs to be simplified. Specifically, it involves combining like terms of a polynomial, which are terms that have the same variable raised to the same power. The expression contains various terms with powers of 'a', including a^6, a^4, a^2, and constant terms without any variables. The task is to perform the algebraic operation of addition by adding the coefficients (numerical parts) of the like terms and combining the constant terms to simplify the expression to its simplest form.
$\left(\right. - 5 a^{4} - 7 a^{6} + 3 \left.\right) + \left(\right. 3 a^{6} + 9 - 9 a^{4} + 5 a^{2} \left.\right)$
Answer
Solution:
Step 1:
Eliminate the parentheses in the expression: $-5a^4 - 7a^6 + 3 + 3a^6 + 9 - 9a^4 + 5a^2$.
Step 2:
Combine like terms by subtracting $9a^4$ from $-5a^4$: $-14a^4 - 7a^6 + 3 + 3a^6 + 9 + 5a^2$.
Step 3:
Combine $-7a^6$ and $3a^6$: $-14a^4 - 4a^6 + 3 + 9 + 5a^2$.
Step 4:
Proceed to simplify the polynomial.
Step 4.1:
Combine the constant terms $3$ and $9$: $-14a^4 - 4a^6 + 12 + 5a^2$.
Step 4.2:
Position the constant term $12$ at the end of the expression: $-14a^4 - 4a^6 + 5a^2 + 12$.
Step 4.3:
Rearrange the terms in descending order of exponents: $-4a^6 - 14a^4 + 5a^2 + 12$.
Knowledge Notes:
To simplify a polynomial expression, follow these steps:
Eliminate Parentheses: Apply the distributive property to remove any parentheses by multiplying any term outside the parentheses by each term inside.
Combine Like Terms: Look for terms that have the same variable raised to the same power. These are like terms and can be combined by adding or subtracting their coefficients.
Simplify Constant Terms: If there are constant terms (numbers without variables), add or subtract them to get a single constant.
Order Terms: It is customary to write polynomial expressions in descending order of their exponents.
Maintain Mathematical Integrity: When combining like terms or rearranging terms, ensure that the signs (+ or -) associated with each term are correctly applied.
In the given problem, the expression contains like terms that can be combined. The process involves careful addition and subtraction of the coefficients of these like terms. The final expression is ordered by the degree of each term, from highest to lowest.
