Write whether the statement is True or False. Justify your answer.
The degree of the sum of two polynomials each of degree $5$ is always $5$.
The degree of the sum of two polynomials each of degree $5$ is always $5$.
(FALSE) The given statement is false.
Justification: The degree of a polynomial is the highest power of the variable in the polynomial.
Consider two polynomials $P(x) = -x^{5} + 3x^{2} + 4$ and $Q(x) = x^{5} + x^{4} + 2x^{3} + 3$.
Both $P(x)$ and $Q(x)$ have a degree of $5$.
Now,find their sum: $P(x) + Q(x) = (-x^{5} + 3x^{2} + 4) + (x^{5} + x^{4} + 2x^{3} + 3) = x^{4} + 2x^{3} + 3x^{2} + 7$.
The degree of the resulting polynomial is $4$,which is not $5$. Thus,the statement is false.
Justification: The degree of a polynomial is the highest power of the variable in the polynomial.
Consider two polynomials $P(x) = -x^{5} + 3x^{2} + 4$ and $Q(x) = x^{5} + x^{4} + 2x^{3} + 3$.
Both $P(x)$ and $Q(x)$ have a degree of $5$.
Now,find their sum: $P(x) + Q(x) = (-x^{5} + 3x^{2} + 4) + (x^{5} + x^{4} + 2x^{3} + 3) = x^{4} + 2x^{3} + 3x^{2} + 7$.
The degree of the resulting polynomial is $4$,which is not $5$. Thus,the statement is false.
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