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Section 1.4 : Polynomials

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

Subtract \(3{\left( {{x^2} + 1} \right)^2}\) from \(6{x^3} - 9{x^2} - 13x - 4\)

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Start Solution

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

\[6{x^3} - 9{x^2} - 13x - 4 - 3{\left( {{x^2} + 1} \right)^2}\]

Now, before we actually do the subtraction we need to actually multiply out the second term before we do the subtraction. Here are the results of all these operations.

\[\begin{align*}6{x^3} - 9{x^2} - 13x - 4 - 3{\left( {{x^2} + 1} \right)^2} & = 6{x^3} - 9{x^2} - 13x - 4 - 3\left( {{x^4} + 2{x^2} + 1} \right)\\ & = 6{x^3} - 9{x^2} - 13x - 4 - 3{x^4} - 6{x^2} - 3\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 3{x^4} + 6{x^3} - 15{x^2} - 13x - 7}}\end{align*}\] 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.