Question

Use a vertical format to add or subtract. $$\left(7 x^{4}-x^{2}+3 x\right)-\left(x^{3}+6 x^{2}-2 x+9\right)$$

Short answer

-\(7x^{4} - x^{3} - 7x^{2} + 5x - 9\)

Step-by-step solution

Rewrite the problem in vertical format

To begin, rearrange the expression in vertical format for ease of calculation, making sure to align the like terms. It should look like this:\[ \begin{align*} & 7x^{4} - x^{2} + 3x \ & - (x^{3} + 6x^{2} - 2x + 9) \end{align*} \]

Distribute the negative sign

Remember, when subtracting polynomials, distribute the negative sign to every term in the second polynomial. The expression should now look like this:\[ \begin{align*} & 7x^{4} - x^{2} + 3x \ & - x^{3} - 6x^{2} + 2x - 9 \end{align*} \]

Combine like terms

The next step is to combine like terms. In this case, \(x^{4}\), \(x^{3}\), \(x^{2}\), \(x^{1}\), and the constant terms are the like terms:\[ \begin{align*} & 7x^{4} + 0x^{3} - x^{2} + 3x + 0 \ & 0x^{4} - x^{3} - 6x^{2} + 2x - 9 \end{align*} \]Combining them gives us \(7x^{4} - x^{3} - 7x^{2} + 5x - 9\)