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Cubic Expansion

By Vighnesh Hemnani

After learning about BEDMAS or PEDMAS (or whichever version you learnt!), you now know how to unravel an equation. But one version of expansion is the cubic expansion. Cubic expansion is an extension of regular expansion except to the third degree. Learn more about its formula, examples and applications.

Why is this concept useful?

Often times the binomial cubic expansion pops up in a lot of real-life applications such as finding the volume of a cube. In those situations, rather than multiplying and expanding out each pair of parentheses, it would be efficient to just apply the formula as written above to save time and energy.

Where does this concept fit into the curriculum?

High School Algebra

What does cubic expansion mean?

Simply put, cubic expansion is expansion of any parentheses containing expressions that are raised to the third degree. However, here, our focus will be only binomial cubic expansions, where the word binomial just means only expansion of exactly two terms inside the parentheses (not one, or three but exactly two!). To be explicit, we will look at cubic expansion that similar to the one below:

(a+b)3= (a+b)(a+b)(a+b)=a3+3a2b+3ab2+b3(a−b)3= (a−b)(a−b)(a−b)=a3−3a2b+3ab2−b3 \begin{array}{l} ( a+b)^{3} =\ ( a+b)( a+b)( a+b) =a^{3} +3a^{2} b+3ab^{2} +b^{3}\\ ( a-b)^{3} =\ ( a-b)( a-b)( a-b) =a^{3} -3a^{2} b+3ab^{2} -b^{3} \end{array}

How to use this concept?

You can apply this concept in a variety of situations when one of the variables inside the parenthesis is unknown and you need the expansion to solve a problem. Or the expansion formula above can be used when a and b are both numbers that when added is big and you can’t find the sum’s cube, but using the formula, you breakdown the expansion in a more bitesize manner allowing you to do the calculations easily.

Sample Math Problems

Question: Use the formula to expand:

(x+y)3( x+y)^{3}

Solution:

x3+3x2y+3xy2+y3x^{3} +3x^{2} y+3xy^{2} +y^{3}

Question: Suppose there is a blacksmith who made a metal cube with the side length of m inches. However, this blacksmith wasn’t too happy with the size of the cube, so he added 3 inches to the side length (to all sides), which would increase the volume the cube takes up. What is the new volume of the cube in terms of m?

Solution: Recall that the volume of a cube is just the side length cubed. So, we know that the previous length was m, but now it is m+3. So, to find the new volume, we just need to apply the formula of a cube:

V=side3V=(m+3)3 \begin{array}{l} V=side^{3}\\ V=( m+3)^{3} \end{array}

Using the cubic expansion formula:

V=m3+3m23+3m32+33V=m3+9m2+27m+27 \begin{array}{l} V=m^{3} +3m^{2} 3+3m3^{2} +3^{3}\\ V=m^{3} +9m^{2} +27m+27 \end{array}

Question: Based on your answer to your previous question, what is the ratio of the new volume to the old volume? That is, by what factor did the volume of the cube increase?

Solution: To do this, it would mean to simply divide the new volume by the old. We already have the new volume and the old volume is simply just m3. Thus:

(m3+9m2+27m+27)m3=m3m3+9m2m3+27mm3+27m3=1+9m+27m2+27m3 \begin{array}{l} \frac{\left( m^{3} +9m^{2} +27m+27\right)}{m^{3}} =\frac{m^{3}}{m^{3}} +\frac{9m^{2}}{m^{3}} +\frac{27m}{m^{3}} +\frac{27}{m^{3}}\\ \\ =1+\frac{9}{m} +\frac{27}{m^{2}} +\frac{27}{m^{3}} \end{array}

Question: Now let’s say the same blacksmith from the previous question with the enlarged cube (with side length m+3), was still not happy with its size and wanted to increase it to side length of m+5. What is the factor by which the volume changed from a side length with m+3 to m+5?

Solution: Similar to our previous approach, we first need to find the new volume with side length m+5, which means all we need to do is cube it:

V=(m+5)3V=( m+5)^{3}

Using the cubic expansion formula:

V=m3+3m25+3m52+53V=m3+15m2+75m+125 \begin{array}{l} V=m^{3} +3m^{2} 5+3m5^{2} +5^{3}\\ V=m^{3} +15m^{2} +75m+125 \end{array}

Therefore, the factor by which the volume increase is:

new volold vol=(m3+15m2+75m+125)(m3+9m2+27m+27)\frac{new\ vol}{old\ vol} =\frac{\left( m^{3} +15m^{2} +75m+125\right)}{\left( m^{3} +9m^{2} +27m+27\right)}



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Practice Math Problems

1. Use the formula to expand the

(d+g)3( d+g)^{3}

2. Use the formula to expand the

(d+3)3( d+3)^{3}

3. Use the formula to expand the

(5+3)3( 5+3)^{3}

4. Suppose a carpenter has a cube with the side length of x inches. However, he needed to increase the size of the cube for a different project, so he added 4 inches to the side length, increasing the volume. Find the new volume of the cube in terms of x.

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