Subtract polynomials: $ \, \color{blue}{ 3x^5+7x^2-4x+8 } \, $ , $ \, \color{orangered}{ 4x^4+16x-3x+7 }$.

$$ \left( \color{blue}{3x^5+7x^2-4x+8} \right) - \left( \color{orangered}{4x^4+16x-3x+7} \right) = 3x^5-4x^4+7x^2-17x+1 $$

Step 1: First we have to write polynomials in descending order.

$$ \begin{aligned} P(x) &= 3x^5+7x^2-4x+8 \\ Q(x) &= 4x^4+13x+7 \\ \end{aligned} $$

Step 2: Write second polynomial below the first one. Making sure like terms are lined up.

$$ \begin{array}{lrrrrr} & 3x^5 & & 7x^2 & -4x & 8 \\ \color{red}{ - } & & 4x^4 & & 13x & 7 \\ \hline \end{array} $$

Step 3: Change ALL the signs in the second row.

$$ \begin{array}{lrrrrr} & 3x^5 & & 7x^2 & -4x & 8 \\ \color{red}{ - } & & -4x^4 & & -13x & -7 \\ \hline \end{array} $$

Step 4: Combine like terms (add each column):

$$ \begin{array}{lrrrrr} & 3x^5 & & 7x^2 & -4x & 8 \\ \color{red}{ - } & & -4x^4 & & -13x & -7 \\ \hline & 3x^5 & 4x^4 & 7x^2 & 9x & 15 \end{array} $$

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