Degree of the polynomial $$x2$$−$$1$$−$$x34+x2+1$$−$$x34$$

We have $$(x$$−$$y)4+(x+y)4=2$$ $$4C0x4+4C2x2y2+4C4y4Thus$$, $$x2$$−$$1$$−$$x34+x2+1$$−$$x34=2$$ $$4C0x24+4C2x221$$−$$x3+4C41$$−$$x32which$$ is a polynomial of degree $$8$$.

Binomial theorem for positive integral Index

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Question

Degree of the polynomial
${(x^{2} - \sqrt{1 - x^{3}})}^{4} + {(x^{2} + \sqrt{1 - x^{3}})}^{4}$

Moderate

Solution

We have
$(x - y)^{4} + (x + y)^{4} = 2 [^{4} C_{0} x^{4} +^{4} C_{2} x^{2} y^{2} +^{4} C_{4} y^{4}]$
Thus, $\begin{array}{l} {(x^{2} - \sqrt{1 - x^{3}})}^{4} + {(x^{2} + \sqrt{1 - x^{3}})}^{4} \\ = 2 [^{4} C_{0} {(x^{2})}^{4} +^{4} C_{2} {(x^{2})}^{2} (1 - x^{3}) +^{4} C_{4} {(1 - x^{3})}^{2}] \end{array}$
which is a polynomial of degree $8 .$

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Binomial theorem for positive integral Index

Remember concepts with our Masterclasses.

Degree of the polynomial x2−1−x34+x2+1−x34

We have (x−y)4+(x+y)4=2 4C0x4+4C2x2y2+4C4y4Thus, x2−1−x34+x2+1−x34=2 4C0x24+4C2x221−x3+4C41−x32which is a polynomial of degree 8.

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Degree of the polynomial
${(x^{2} - \sqrt{1 - x^{3}})}^{4} + {(x^{2} + \sqrt{1 - x^{3}})}^{4}$