# Determine how many linear factors and zeros each polynomial function has. P(x)=x^{40} + x^{39}

Determine how many linear factors and zeros each polynomial function has.
$P\left(x\right)={x}^{40}+{x}^{39}$
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mhalmantus
Number of zeros theorem:
If multiple zeros are counted individually, the polynomial function P(x) with degree $n\left(n>0\right)$
has exactly n zeros among the complex numbers.
The Polynomial Factorization Theorem:
If $n>0$ and P(x) is an nth-degree polynomial function, then P(x) has exactly n linear factors:
$P\left(x\right)={a}_{n}\left(x-{c}_{1}\right)\left(X-{c}_{2}\right)\left(X-{c}_{3}\right).....\left(x-{C}_{n}\right)$
Where ${c}_{1},{c}_{2},{c}_{3},....,{c}_{n}$ are numbers and ${a}_{n}$, is the leading coefficient of P(x).
Finding zeros:
Given:
$P\left(x\right)={x}^{40}+{x}^{39}$
According to the Number of zeros theorem, the polynomial function P(x) has n zeros.
Where n is the degree of the polynomial function
Here $n=40$
Hence the zeros of the polynomial function $P\left(x\right)={x}^{40}+{x}^{39}$ is 40
Finding linear factors:
Given:
$P\left(x\right)={x}^{40}+{x}^{39}$
According to The Polynomial Factorization Theorem, the polynomial function P(x) has n linear factors.
Where n is the degree of the polynomial function
Here $n=40$
Hence the linear factors of the polynomial function $P\left(x\right)={x}^{40}+{x}^{39}$ is 40
Final statement:
The number of linear factors of the polynomial function $P\left(x\right)={x}^{40}+{x}^{39}$ are 40.
The zeros of the polynomial function $P\left(x\right)={x}^{40}+{x}^{39}$ are 40.