Questions: Find the zeros of the function and state the multiplicities. m(x) = x^6 - 7x^4 If there is more than one answer, separate them with commas. Select "None" if applicable.

Find the zeros of the function and state the multiplicities.
m(x) = x^6 - 7x^4

If there is more than one answer, separate them with commas. Select "None" if applicable.

Transcript text: Find the zeros of the function and state the multiplicities. \[ m(x)=x^{6}-7 x^{4} \] If there is more than one answer, separate them with commas. Select "None" if applicable.

Solution

Solution Steps

Step 1: Factorize the Polynomial

Attempt to factorize the polynomial into its simplest form. This involves identifying any common factors or special factorization formulas applicable to the polynomial.

Step 2: Use the Rational Root Theorem

For polynomials with integer coefficients, identify possible rational roots that can be tested.

Step 3: Apply Synthetic Division or Polynomial Division

Test potential roots and further simplify the polynomial using division techniques.

Step 4: Solve the Factored Equations

Set each factor equal to zero and solve for the variable to find the zeros of the polynomial.

Step 5: Multiplicity

Determine the multiplicity of each root by observing the power of the factor in the polynomial.

Step 6: Complex Roots

Consider that there may be complex roots, especially for polynomials of degree greater than 2. The Fundamental Theorem of Algebra assures us that a polynomial of degree \(n\) has exactly \(n\) roots in the complex number system, counting multiplicities.

Step 7: Numerical Methods

For higher-degree polynomials where analytical solutions are not feasible, numerical methods such as Newton's method or computational tools can be employed to approximate the roots.