Question

use a vertical format to subtract the polynomials.
\\(\

$$\begin{array}{r} -0.1y^{19} + 0.2y^{8} + 0.9y^{7} - 0.8\\\\ -(-0.4y^{19} + 0.5y^{8} - 0.7y^{7} - 0.8)\\\\ \\hline \\square \\end{array}$$

\\)

Explanation:

Step1: Distribute the negative sign

When we subtract a polynomial, we distribute the negative sign to each term inside the parentheses. So, \(-(-0.4y^{19}+0.5y^{8}-0.7y^{7}-0.8)\) becomes \(+0.4y^{19}-0.5y^{8}+0.7y^{7}+0.8\). Now our polynomial subtraction in vertical format is:

$$
LATEXBLOCK0
$$

Step2: Combine like terms for \(y^{19}\)

The coefficients of \(y^{19}\) are \(- 0.1\) and \(0.4\). Adding them together: \(-0.1 + 0.4=0.3\). So the term for \(y^{19}\) is \(0.3y^{19}\).

Step3: Combine like terms for \(y^{8}\)

The coefficients of \(y^{8}\) are \(0.2\) and \(-0.5\). Adding them together: \(0.2- 0.5=-0.3\). So the term for \(y^{8}\) is \(-0.3y^{8}\).

Step4: Combine like terms for \(y^{7}\)

The coefficients of \(y^{7}\) are \(0.9\) and \(0.7\). Adding them together: \(0.9 + 0.7 = 1.6\). So the term for \(y^{7}\) is \(1.6y^{7}\).

Step5: Combine like terms for the constant term

The constant terms are \(-0.8\) and \(0.8\). Adding them together: \(-0.8+0.8 = 0\).

Answer:

\(0.3y^{19}-0.3y^{8}+1.6y^{7}\)