#### Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. -Each antiderivative of an nth-degree polynomial function is an (n+1)th-degree polynomial function.

#### Solution

Verified#### Step 1

1 of 2Assume we have the polynomial of degree $n$ which can be written as

$P_n=a_n\ x^n+a_{n-1}\ x^{n-1}+a_{n-2}\ x^{n-2}+\dots+a_1\ x+ a_0$

and now calculate its anti-derivative using sum and power rules

$\int P_n\ dx=a_n\ \frac{x^{n+1}}{n+1}+a_{n-1}\ \frac{x^{n}}{n}+a_{n-2}\ \frac{x^{n-1}}{n-1}+\dots+a_1\ \frac{x^2}2+ a_0x+C$

which can be written as follows

$P_{n+1}\ dx=b_{n+1}\ x^{n+1}+b_{n}\ x^{n}+b_{n-1}\ x^{n-1}+\dots+b_2\ x^2+ b_1\ x+b_0$

Hence the statement is true

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