Simplify (12x^5-3x^4+2x-5)+(8x^4-3x^3+4x+1)
The given problem is a question about algebraic expression simplification. Specifically, the task is to combine two polynomial expressions by performing addition. Each polynomial consists of terms that are combinations of variables raised to an exponent (which are called the degree of the term) and coefficients (which are the numerical multipliers of the terms). To simplify the given expression, you must combine like terms from both polynomials, which are terms that have the same variable raised to the same power. This simplification will result in a new polynomial with combined like terms, each with their respective coefficients adjusted by the addition operation.
$\left(\right. 12 x^{5} - 3 x^{4} + 2 x - 5 \left.\right) + \left(\right. 8 x^{4} - 3 x^{3} + 4 x + 1 \left.\right)$
Answer
Solution:
Step 1:
Eliminate the parentheses to combine like terms: $12x^5 - 3x^4 + 2x - 5 + 8x^4 - 3x^3 + 4x + 1$.
Step 2:
Combine the coefficients of $x^4$: $12x^5 + (8x^4 - 3x^4) + 2x - 5 - 3x^3 + 4x + 1$.
Step 3:
Combine the coefficients of $x$: $12x^5 + 5x^4 + (4x + 2x) - 5 - 3x^3 + 1$.
Step 4:
Consolidate the constant terms and rearrange the polynomial.
Step 4.1:
Combine the constants: $12x^5 + 5x^4 - 3x^3 + 6x + (1 - 5)$.
Step 4.2:
Final arrangement of terms: $12x^5 + 5x^4 - 3x^3 + 6x - 4$.
Knowledge Notes:
The problem involves simplifying a polynomial expression by combining like terms. Here are the relevant knowledge points:
Polynomials: A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.
Like Terms: In algebra, like terms are terms that have the same variables and powers. The coefficients do not need to be the same.
Combining Like Terms: To simplify an expression, you combine like terms by adding or subtracting the coefficients.
Order of Terms: When writing the final expression, it is common to arrange the terms in descending order of their degree (the exponent on the variable).
Parentheses Removal: When you remove parentheses, you must pay attention to the sign in front of them. If there is a subtraction sign before the parentheses, you must change the sign of each term inside when removing the parentheses.
Constants: These are terms without variables and are also combined by addition or subtraction.
In LaTeX, we use the caret (^) symbol for exponentiation, such as $x^2$ for \(x^2\). The backslash (\) is used to indicate a LaTeX command, such as \sum for \(\sum\). Curly braces ({}) group parts of the command that need to be treated as a single unit, and the dollar sign ($) is used to start and end math mode.
