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Section 1.4 : Polynomials

3. Perform the indicated operation and identify the degree of the result.

Subtract \( - 3{x^2} + 7x + 8\) from \({x^4} + 7{x^3} - 12x - 1\)

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Start Solution

Here is the operation we’re being asked to perform.

\[{x^4} + 7{x^3} - 12x - 1 - \left( { - 3{x^2} + 7x + 8} \right)\]

Be careful with the order here! We are subtracting the first polynomial from the second and that implies the order we’ve got here. Also be careful with the parenthesis on the second polynomial. We are subtracting the whole polynomial and so we need to have the parenthesis to do that.

Here’s the result of the operation.

\[\begin{align*}{x^4} + 7{x^3} - 12x - 1 - \left( { - 3{x^2} + 7x + 8} \right) & = {x^4} + 7{x^3} - 12x - 1 + 3{x^2} - 7x - 8\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{{x^4} + 7{x^3} + 3{x^2} - 19x - 9}}\end{align*}\] Show Step 2

Remember the degree of a polynomial is just the largest exponent in the polynomial and so the degree of the result of this operation is four.