91Ó°ÊÓ

A polynomial is a mathematical expression consisting of variables, coefficients, and constants. It is composed of terms, where each term is a product of a coefficient and a variable raised to a non-negative integer power. A simple example of a polynomial is \(P(x) = 3x^2 + 5x - 2\).

Like terms are terms that have the same variable raised to the same power. In the process of subtracting polynomials, it is crucial to identify and match the like terms from both polynomials. For example, if we have two polynomials \(P(x) = 3x^2 - 5x + 1\) and \(Q(x) = -2x^2 + 3x - 4\), the like terms can be paired as follows: From \(P(x)\): \(3x^2\), \(-5x\), and \(1\) From \(Q(x)\): \(-2x^2\), \(3x\), and \(-4\)

To subtract two polynomials, follow these steps: 1. Write down the given polynomials, with the first one at the top and the second one just below it, ensuring that the like terms are aligned vertically. 2. Place a horizontal line below the second polynomial and a minus sign to the left of it, indicating subtraction. 3. Subtract the corresponding terms vertically from the aligned polynomial, and write the result below the line. Make sure to combine the like terms properly. 4. Write down the final resulting polynomial with all the subtracted terms. Using the example polynomials \(P(x)\) and \(Q(x)\) mentioned earlier, we will perform the subtraction: Step 1: Align the like terms. \[ \begin{array}{c@{}c@{}c@{}c@{}c} & 3x^2 & -5x & +1 & \\ - & (-2x^2) & +3x & -4 & \\ \cline{1-5} \end{array} \] Step 2: Subtract the corresponding terms. \[ \begin{array}{c@{}c@{}c@{}c@{}c} & 3x^2 & -5x & +1 & \\ - & (-2x^2) & +3x & -4 & \\ \cline{1-5} & (3x^2 -(-2x^2)) & (-5x-3x) & (1-(-4)) & \\ \end{array} \] Step 3: Write down the final resulting polynomial. \[ \begin{array}{c@{}c@{}c@{}c} 5x^2 & -8x & +5 & \\ \end{array} \] So, the resulting polynomial after subtraction is: \(R(x) = 5x^2 - 8x + 5\).