Harlen Pritchard

2020-10-28

Determine how many linear factors and zeros each polynomial function has.

$P(x)=4{x}^{5}+8{x}^{3}$

crocolylec

Skilled2020-10-29Added 100 answers

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$n>O$ and P(x) is an nth-degree polynomial function, then P(x) has exactly n linear factors:

$P(X)={a}_{n}(x\u2014{c}_{1})(x\u2014{x}_{2})(X-{c}_{3})....(x-{C}_{n})$

Where c1,c2,c3,..... Cn are numbers and a_{n}, is the leading coefficient of P(x).

Finding zeros:

Given:

$P(x)=4{x}^{5}+8{x}^{3}$

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$P(x)=4{x}^{5}+8{x}^{3}$ is 5

Finding linear factors:

Given:

$P(x)=4{x}^{5}+8{x}^{3}$

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$P(x)=4{x}^{5}+8{x}^{3}$ is 5

Final statement:

The number of linear factors of the polynomial function$P(x)=4{x}^{5}+8{x}^{3}$ are 5.

The zeros of the polynomial function$P(x)=4{x}^{5}+8{x}^{3}$ are 5.

If multiple zeros are counted individually, the polynomial function P(x) with degree

The Polynomial Factorization Theorem:

If

Where c1,c2,c3,..... Cn are numbers and a_{n}, is the leading coefficient of P(x).

Finding zeros:

Given:

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

Hence the zeros of the polynomial function

Finding linear factors:

Given:

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

Hence the linear factors of the polynomial function

Final statement:

The number of linear factors of the polynomial function

The zeros of the polynomial function

$\frac{20b}{{\left(4{b}^{3}\right)}^{3}}$

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