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

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

2021-02-28
Number of zeros theorem:
If multiple zeros are counted individually, the polynomial function P(x) with degree $$n(n > 0)$$
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:
$$P(x) = a_{n}(x - c_{1})(X - c_{2})(X - c_{3}).....(x-C_{n})$$
Where $$c_{1},c_{2},c_{3},....,c_{n}$$ are numbers and $$a_{n}$$, is the leading coefficient of P(x).
Finding zeros:
Given:
$$P(x)=x^{40}+x^{39}$$
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 $$P(x) = x^{40} + x^{39}$$ is 40
Finding linear factors:
Given:
$$P(x)=x^{40} +x^{39}$$
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 $$P(x) = x^{40} + x^{39}$$ is 40
Final statement:
The number of linear factors of the polynomial function $$P(x) = x^{40} + x^{39}$$ are 40.
The zeros of the polynomial function $$P(x) = x^{40} + x^{39}$$ are 40.

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