Emeli Hagan

2021-02-27

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

mhalmantus

Amount of zeros theorem: The degree three polynomial function P(x) holds true if numerous zeros are counted independently $n\left(n>0\right)$ has exactly n zeros among the complex numbers.
The Polynomial Factorization Theorem:
If $n>0$ if P(x) has exactly n linear elements and P(x) is a polynomial function of the nth degree:
$P\left(x\right)={a}_{n}\left(x-{c}_{1}\right)\left(X-{c}_{2}\right)\left(X-{c}_{3}\right).....\left(x-{C}_{n}\right)$
Where ${c}_{1},{c}_{2},{c}_{3},....,{c}_{n}$ are numbers and ${a}_{n}$, is the leading coefficient of P(x).
Finding zeros:
Given:
$P\left(x\right)={x}^{40}+{x}^{39}$
The polynomial function P(x) has n zeros, according to the Number of Zeros Theorem.
where n is the polynomial function's degree
Here $n=40$
Hence the zeros of the polynomial function $P\left(x\right)={x}^{40}+{x}^{39}$ is 40
Finding linear factors:
Given:
$P\left(x\right)={x}^{40}+{x}^{39}$
The polynomial function P(x) has n linear factors, according to The Polynomial Factorization Theorem.
where n is the polynomial function's degree
Here $n=40$
Hence the linear factors of the polynomial function $P\left(x\right)={x}^{40}+{x}^{39}$ is 40
The polynomial function's number of linear elements is the conclusion.

$P\left(x\right)={x}^{40}+{x}^{39}$ are 40.
The zeros of the polynomial function $P\left(x\right)={x}^{40}+{x}^{39}$ are 40.

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