High School

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Factor each of these polynomials:

a) [tex]6x^3 + 47x^2 + 71x - 70[/tex]

b) [tex]12x^6 - 56x^5 + 100x^4 - 80x^3 + 20x^2 + 8x - 4[/tex]

Answer :

Final answer:

Factoring polynomials involves expressing them as a product of their factors. Using the difference of cubes formula, the polynomial x³ - y³ can be factored into (x - y)(x² + xy + y²) using both real and complex coefficients.

Explanation:

The student's question involves factoring polynomials, which is a mathematical process used to rewrite a polynomial as a product of its factors. This can simplify the expression and make it easier to work with in equations and other mathematical contexts. To provide an example, let us factor the polynomial x³ - y³:

  • Using complex coefficients: we can factor it using the difference of cubes formula, x³ - y³ = (x - y)(x² + xy + y²).
  • Using real coefficients: we again use the difference of cubes, remembering that the quadratic term cannot be factored further over the reals.

When factoring polynomials, it's also important to look for common factors, the use of various factoring formulas (like difference of squares or difference of cubes), and techniques such as grouping. In some cases, polynomials may also be factored using methods like synthetic division or by finding the zeroes of the polynomial.

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