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.

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.