Understanding Fourth Proportional in Algebra

The concept of proportion involves equating two ratios. When we say four quantities, say \(a\), \(b\), \(c\), and \(d\), are in proportion, it means that the ratio of the first to the second is equal to the ratio of the third to the fourth. Mathematically, this is written as:

In this relationship, \(d\) is called the fourth proportional to \(a\), \(b\), and \(c\).

We are given three terms and asked to find the fourth proportional. Let the three given terms be \(T_1\), \(T_2\), and \(T_3\), and let the fourth proportional be \(x\). According to the definition of proportion, we have:

We can rearrange this equation to solve for the fourth proportional \(x\):

Now, let's identify the given terms:

• \(T_1 = (a + b)^2 \times (a^3 - b^3)\)
• \(T_2 = (a^3 + b^3)\)
• \(T_3 = (a^2 - b^2)\)

Substitute these terms into the formula for the fourth proportional \(x\):

To simplify this expression, we need to use algebraic factorization identities. The relevant identities are:

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

Now, let's substitute these factorized forms into the expression for \(x\):

Let's write out the numerator and denominator factors separately to see which terms can be cancelled:

Numerator factors: \((a + b), (a^2 - ab + b^2), (a - b), (a + b)\)

Denominator factors: \((a + b), (a + b), (a - b), (a^2 + ab + b^2)\)

We can cancel terms that appear in both the numerator and the denominator. Notice that \((a+b)\) appears twice in the numerator and twice in the denominator. Also, \((a-b)\) appears once in both.

Cancelling these common factors:

• One \((a+b)\) from the numerator cancels with one \((a+b)\) from the denominator.
• The second \((a+b)\) from the numerator cancels with the second \((a+b)\) from the denominator.
• The \((a-b)\) from the numerator cancels with the \((a-b)\) from the denominator.

After cancellation, the remaining terms are:

Numerator remaining: \((a^2 - ab + b^2)\)

Denominator remaining: \((a^2 + ab + b^2)\)

So, the simplified expression for the fourth proportional \(x\) is:

This matches one of the given options.

Revision Table: Key Algebraic Identities

Additional Information on Proportion Concepts

Proportion is a fundamental concept in mathematics, used to show that two ratios are equal. Besides the fourth proportional, there are other related terms:

• Third Proportional: If \(a, b, c\) are in continued proportion, then \(\frac{a}{b} = \frac{b}{c}\). Here, \(c\) is the third proportional to \(a\) and \(b\).
• Mean Proportional: If \(a, b, c\) are in continued proportion \(\frac{a}{b} = \frac{b}{c}\), then \(b\) is the mean proportional between \(a\) and \(c\). From the equation, \(b^2 = ac\), so \(b = \sqrt{ac}\).

Understanding how to factorize algebraic expressions is crucial for solving problems involving proportions with polynomials, as demonstrated in finding the fourth proportional in this problem.