Factoring Polynomials: A Step-by-Step Guide with Google SEO Strategies

When looking to factor a polynomial like x^3 - 3x^2 - 24x - 80, several mathematical techniques come into play. This guide will walk you through the process using the Rational Root Theorem and polynomial long division. Additionally, we'll discuss how to optimize this content for Google's search engines.

Introduction to Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra. It involves breaking down a polynomial into simpler expressions that, when multiplied together, give the original polynomial. The given polynomial is x^3 - 3x^2 - 24x - 80. To factor this, one can use the Rational Root Theorem, which helps identify possible rational roots.

Rational Root Theorem and Possible Roots

The Rational Root Theorem states that any rational root, expressed as a fraction (frac{p}{q}), of the polynomial (a_nx^n a_{n-1}x^{n-1} ... a_1x a_0 0) must have (frac{p}{q}) as a factor of the constant term (a_0) and the leading coefficient (a_n).

For the polynomial x^3 - 3x^2 - 24x - 80, the possible rational roots are the factors of the constant term 80 (±1, ±2, ±4, ±5, ±8, ±10, ±16, ±20, ±40, ±80) divided by the leading coefficient 1. Therefore, the possible rational roots are ±1, ±2, ±4, ±5, ±8, ±10, ±16, ±20, ±40, ±80.

Testing Possible Roots

To find the actual roots, we can test these values. Let's test x 4.

:math:`4^3 - 3(4^2) - 24(4) - 80 64 - 48 - 96 - 80 0`

Thus, x 4 is a root.

Knowing that x - 4 is a factor, we can divide the polynomial x^3 - 3x^2 - 24x - 80 by x - 4 using polynomial long division.

Polynomial Long Division

Step-by-Step Division:

Divide x^3 by x to get x^2. Multiply x^2 by x - 4 to get x^3 - 4x^2. Subtract x^3 - 4x^2 from x^3 - 3x^2 - 24x - 80 to get x^2 - 24x - 80). Divide x^2 by x to get x. Multiply x by x - 4 to get x^2 - 4x. Subtract x^2 - 4x - 80 from x^2 - 24x - 80 to get -24x - 80. Divide -24x - 80 by x to get -20. Multiply -20 by x - 4 to get -2 80. Subtract -2 80 from -24x - 80 to get 0.

The result of the division is x^2 x - 20.

Factoring the Result

Next, we factor x^2 x - 20.

x^2 x - 20 (x 5)(x - 4)

Complete Factorization

Therefore, the complete factorization of the polynomial x^3 - 3x^2 - 24x - 80 is:

(x - 4)(x 5)(x - 4) (x - 4)^2(x 5)

The factors are:

x - 4 with multiplicity 2 x 5

SEO Optimization for Google

To optimize this content for Google, we need to use SEO techniques. Here are some strategies:

Keyword Optimization: Use the keywords polynomial factoring, rational root theorem, and google SEO throughout the text, especially in headings, subheadings, and the content. Title Tag: Ensure the title tag is descriptive and includes the main keywords, e.g., "How to Factor Polynomials Using the Rational Root Theorem and Polynomial Long Division." Use H1 tags for the main heading and H2 for subheadings. Meta Description: Write a concise meta description that includes the main keywords and provides a brief summary of the content. Internal Linking: Link to other relevant pages on your site that discuss factoring polynomials, rational root theorem, or polynomial long division. External Linking: Link to reputable sources or resources that provide more information about these topics for reference. Image Optimization: Include relevant images and use descriptive alt text that includes the main keywords. Content Length: Aim for longer, detailed content (3000 words) as Google prefers content that provides in-depth information. Mobile-Friendly Design: Ensure the content is easy to read on mobile devices, as Google gives mobile-friendly sites a preference in its search rankings. Backlinks: Encourage other websites to link back to your content, as backlinks are a key ranking factor for Google.

Conclusion

In conclusion, the polynomial x^3 - 3x^2 - 24x - 80 can be factored as (x - 4)^2(x 5). Optimizing this content for Google involves using appropriate keywords, structuring content with headings, and implementing optimization strategies.