 # Determine how many linear factors and zeros each polynomial function has. P(x)=4x^{5}+8x^{3} Harlen Pritchard 2020-10-28 Answered
Determine how many linear factors and zeros each polynomial function has.
$P\left(x\right)=4{x}^{5}+8{x}^{3}$
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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 $n>O$ and P(x) is an nth-degree polynomial function, then P(x) has exactly n linear factors:
$P\left(X\right)={a}_{n}\left(x—{c}_{1}\right)\left(x—{x}_{2}\right)\left(X-{c}_{3}\right)....\left(x-{C}_{n}\right)$
Where c1,c2,c3,..... Cn are numbers and a_{n}, is the leading coefficient of P(x).
Finding zeros:
Given:
$P\left(x\right)=4{x}^{5}+8{x}^{3}$
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 $n=5$
Hence the zeros of the polynomial function $P\left(x\right)=4{x}^{5}+8{x}^{3}$ is 5
Finding linear factors:
Given:
$P\left(x\right)=4{x}^{5}+8{x}^{3}$
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 $n=5$
Hence the linear factors of the polynomial function $P\left(x\right)=4{x}^{5}+8{x}^{3}$ is 5
Final statement:
The number of linear factors of the polynomial function $P\left(x\right)=4{x}^{5}+8{x}^{3}$ are 5.
The zeros of the polynomial function $P\left(x\right)=4{x}^{5}+8{x}^{3}$ are 5.