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Section 1.4 : Polynomials

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

\[3{\left( {10 - 4{y^3}} \right)^2}\]

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Remember that this is just another way of writing,

\[3{\left( {10 - 4{y^3}} \right)^2} = 3\left( {10 - 4{y^3}} \right)\left( {10 - 4{y^3}} \right)\]

Now all we need to do is foil out the two polynomials. Here is the result of doing that.

\[3{\left( {10 - 4{y^3}} \right)^2} = 3\left( {10 - 4{y^3}} \right)\left( {10 - 4{y^3}} \right) = 3\left( {100 - 80{y^3} + 16{y^6}} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{300 - 240{y^3} + 48{y^6}}}\]

Be careful with dealing with the three! Make sure you take care of the exponent first (i.e. make sure you multiply out the product first) before you multiply the three through the result.

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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 six.