Polynomials are fundamental mathematical expressions that involve variables raised to various powers. Subtraction of polynomials involves simplifying and determining the difference between two polynomial expressions. In this article, we will explore and compute the difference between the given polynomials: (5x^3 + 4x^2) and (6x^2 – 2x – 9). By understanding the concept of polynomial subtraction and applying the appropriate operations, we can determine the difference between these expressions.
The Given Polynomials
(5x^3 + 4x^2) – (6x^2 – 2x – 9)
Simplifying the Subtraction
To subtract polynomials, we distribute the negative sign to each term within the parentheses of the second polynomial. This can be done by changing the signs of each term as follows:
(5x^3 + 4x^2) – 6x^2 + 2x + 9
Combining Like Terms
Next, we combine the like terms within the expression by adding or subtracting coefficients of the same variables raised to the same powers. In this case, we focus on the terms with x^3, x^2, x, and the constant term.
For x^3: There is only one term with x^3 in the expression, which is 5x^3. So, we leave it as is.
For x^2: We have 4x^2 in the first polynomial and -6x^2 in the second polynomial. To find the difference, we subtract the coefficients: 4x^2 – 6x^2 = -2x^2.
For x: We have no terms with x in the first polynomial. In the second polynomial, we have 2x. Since there are no terms to combine, we simply write it as 2x.
For the constant term: In the first polynomial, there is no constant term. In the second polynomial, we have -9. So, we write it as -9.
Combining all the terms, the simplified difference is:
5x^3 – 2x^2 + 2x – 9
The difference between the polynomials (5x^3 + 4x^2) and (6x^2 – 2x – 9) is represented by the expression 5x^3 – 2x^2 + 2x – 9. By distributing the negative sign and combining like terms, we can simplify the expression to accurately represent the difference between the given polynomials. Polynomial subtraction involves careful attention to the powers of variables and the corresponding coefficients. Understanding these operations allows us to perform computations and manipulate polynomial expressions effectively.