Simplify (x^2-3x^3+6x-2)+(5x-4x^2-4)
You are tasked with performing polynomial addition. This involves combining like terms, which are terms in a polynomial with the same variables raised to the same powers, from two separate polynomials. Specifically, the problem presents two polynomials, each with multiple terms involving the variable x raised to different powers. You must add these polynomials by summing the coefficients (numerical factors) of like terms, which will simplify the expression into a single polynomial.
$\left(\right. x^{2} - 3 x^{3} + 6 x - 2 \left.\right) + \left(\right. 5 x - 4 x^{2} - 4 \left.\right)$
Answer
Solution:
Step 1:
Eliminate the parentheses to combine like terms: $x^2 - 3x^3 + 6x - 2 + 5x - 4x^2 - 4$.
Step 2:
Combine the $x^2$ terms by subtracting $4x^2$ from $x^2$: $-3x^3 - 3x^2 + 6x - 2 + 5x - 4$.
Step 3:
Combine the $x$ terms by adding $6x$ to $5x$: $-3x^3 - 3x^2 + 11x - 2 - 4$.
Step 4:
Final simplification of the expression.
Step 4.1:
Combine the constant terms by subtracting $4$ from $-2$: $-3x^3 - 3x^2 + 11x - 6$.
Step 4.2:
Rearrange the terms in descending order of powers of $x$: $-3x^3 - 3x^2 + 11x - 6$.
Knowledge Notes:
The problem involves simplifying a polynomial expression by combining like terms. The process entails the following knowledge points:
Polynomial Expressions: A polynomial is an expression consisting of variables (also known as indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.
Combining Like Terms: Like terms are terms that contain the same variables raised to the same power. Only the coefficients of like terms can be added or subtracted.
Order of Operations: When simplifying expressions, it is important to follow the order of operations, which typically is parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).
Descending Powers: When writing the final expression, it is common practice to arrange the terms in descending order of their powers.
LaTeX Formatting: In the solution, LaTeX is used to properly format mathematical expressions, allowing for clear and precise presentation of equations and formulas.
In this problem, the steps involve removing parentheses, combining like terms, and rearranging the expression to present it in a simplified form.
