Question

a) Factor each polynomial. i) \(x^{3}-1\) ii) \(x^{3}-27\) iii) \(x^{3}+1\) iv) \(x^{3}+64\) b) Use the results from part a) to decide whether \(x+y\) or \(x-y\) is a factor of \(x^{3}+y^{3} .\) State the other factor(s). c) Use the results from part a) to decide whether \(x+y\) or \(x-y\) is a factor of \(x^{3}-y^{3} .\) State the other factor(s). d) Use your findings to factor \(x^{6}+y^{6}\)

Short answer

a) \(x^{3}-1 = (x-1)(x^{2}+x+1)\), \(x^{3}-27 = (x-3)(x^{2}+3x+9)\), \(x^{3}+1 = (x+1)(x^{2}-x+1)\), \(x^{3}+64 = (x+4)(x^{2}-4x+16)\). b) \(x+y\) is a factor of \(x^{3}+y^{3}\), other factors are \(x^{2}-xy+y^{2}\). c) \(x-y\) is a factor of \(x^{3}-y^{3}\), other factors are \(x^{2}+xy+y^{2}\).

Step-by-step solution

Factor the polynomial for Part a)

Consider the polynomials one by one. i) For the polynomial \(x^{3}-1\), apply the difference of cubes formula \(a^{3}-b^{3} = (a-b)(a^{2}+ab+b^{2})\). Here, \(a = x\) and \(b = 1\).\[(x-1)(x^{2}+x+1)\]ii) For the polynomial \(x^{3}-27\), use the same formula with \(a = x\) and \(b = 3\).\[(x-3)(x^{2}+3x+9)\]iii) For the polynomial \(x^{3}+1\), use the sum of cubes formula \(a^{3}+b^{3} = (a+b)(a^{2}-ab+b^{2})\). Here, \(a = x\) and \(b = 1\).\[(x+1)(x^{2}-x+1)\]iv) For the polynomial \(x^{3}+64\), apply the sum of cubes formula with \(a = x\) and \(b = 4\).\[(x+4)(x^{2}-4x+16)\]

Determine factors of \(x^{3}+y^{3}\)

For \(x^{3}+y^{3}\), use the sum of cubes formula \(a^{3}+b^{3} = (a+b)(a^{2}-ab+b^{2})\) with \(a = x\) and \(b = y\). Thus, \(x+y\) is a factor.\[(x+y)(x^{2}-xy+y^{2})\]The other factor is \(x^{2}-xy+y^{2}\).

Determine factors of \(x^{3}-y^{3}\)

For \(x^{3}-y^{3}\), use the difference of cubes formula \(a^{3}-b^{3} = (a-b)(a^{2}+ab+b^{2})\) with \(a = x\) and \(b = y\). Thus, \(x-y\) is a factor.\[(x-y)(x^{2}+xy+y^{2})\]The other factor is \(x^{2}+xy+y^{2}\).

Factor \(x^{6}+y^{6}\)

Rewrite \(x^{6}+y^{6}\) as \((x^{3})^{2}+(y^{3})^{2}\). This is a sum of squares, which can sometimes be factored further, but not using simple formulas like sum of cubes or difference of cubes. Therefore, the expression is in its simplest form without real factors.

Key concepts

Difference of Cubes

The difference of cubes formula is a special factoring method used for polynomials of the form \(a^3 - b^3\). This formula states that \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).
To see how this works, consider the polynomial \(x^3 - 1\). Here, we recognize it as a difference of cubes where \(a = x\) and \(b = 1\). Applying the formula, we get:\[(x - 1)(x^2 + x + 1)\] This method is very useful and can be applied to other polynomials that fit the difference of cubes format, like \(x^3 - 27\). For this case, \(a = x\) and \(b = 3\), leading to:\[(x - 3)(x^2 + 3x + 9)\] Understanding this pattern simplifies the process of factoring polynomials significantly.

Sum of Cubes

The sum of cubes formula is another essential tool for factoring specific polynomials. This formula states that \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
Let's look at the polynomial \(x^3 + 1\). It's a sum of cubes where \(a = x\) and \(b = 1\). Applying the sum of cubes formula, we obtain:\[(x + 1)(x^2 - x + 1)\] Similarly, for \(x^3 + 64\), identifying it as a sum of cubes with \(a = x\) and \(b = 4\), we get:\[(x + 4)(x^2 - 4x + 16)\] This formula helps to quickly and accurately factor polynomials in the sum of cubes form. Practicing identification and application of this formula makes polynomial factoring much easier.

Polynomial Identities

Polynomial identities are important tools in algebra that provide a framework for understanding how polynomials behave and how they can be manipulated or factored.
Key polynomial identities include the difference of cubes and sum of cubes. These identities provide specific patterns for factoring:

• Difference of cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
• Sum of cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)

These patterns help streamline the factoring process by providing ready-made formulas. Taking advantage of these identities can simplify many algebraic tasks. It's also important to note the symmetrical nature of these formulas, making them easy to remember once familiar.

Factoring Polynomials

Factoring polynomials is a core skill in algebra that involves breaking down a polynomial into simpler, multiplied factors.
Techniques such as applying the difference of cubes and sum of cubes formulas are particularly helpful:

• For \(x^3 - y^3\), using the difference of cubes formula provides factors \(x - y\) and \(x^2 + xy + y^2\).
• For \(x^3 + y^3\), applying the sum of cubes formula gives factors \(x + y\) and \(x^2 - xy + y^2\).

Beyond these specific types, there are other factoring techniques like finding greatest common factors or using quadratic methods. Each polynomial presents unique challenges, and learning various factoring methods is crucial to addressing them effectively. Factoring is not only about finding factors; it also reveals inherent structures and symmetries within the polynomial, aiding in deeper mathematical understanding.