Processing math: 100%

Subscribe to our YouTube channel for the latest videos, updates, and tips.


Cube of a Binomial

How do you get the cube of a binomial?

For cubing a binomial we need to know the formulas for the sum of cubes and the difference of cubes.

Sum of cubes:

The sum of a cubed of two binomial is equal to the cube of the first term, plus three times the square of the first term by the second term, plus three times the first term by the square of the second term, plus the cube of the second term.

(a + b)3 = a3 + 3a2b + 3ab2 + b3

            = a3 + 3ab (a + b) + b3

Difference of cubes:     

The difference of a cubed of two binomial is equal to the cube of the first term, minus three times the square of the first term by the second term, plus three times the first term by the square of the second term, minus the cube of the second term.



(a – b)3 = a3 – 3a2b + 3ab2 – b3

            = a3 – 3ab (a – b) – b3


Worked-out examples for the expansion of cube of a binomial:

Simplify the following by cubing:

1. (x + 5y)3 + (x – 5y)3

Solution:

We know, (a + b)3 = a3 + 3a2b + 3ab2 + b3

and,

(a – b)3 = a3 – 3a2b + 3ab2 – b3

Here, a = x and b = 5y

Now using the formulas for cube of two binomials we get,

= x3 + 3.x2.5y + 3.x.(5y)2 + (5y)3 + x3 - 3.x2.5y + 3.x.(5y)2 - (5y)3

= x3 + 15x2y + 75xy2 + 125 y3 + x3 - 15x2y + 75xy2 - 125 y3

= 2x3 + 150xy2

Therefore, (x + 5y)3 + (x – 5y)3 = 2x3 + 150xy2


2. (12x+32y)3+(12x32y)3

Solution:

Here a = 12x,b=32y

 =(12x)3+3(12x)232y+312x(32y)2+(32y)3+(12x)33(12x)232y+312x(32y)2(32y)3

 =18x3+98x2y+278xy2+278y3+18x398x2y+278xy2278y3

 =18x3+18x3+278xy2+278xy2

 =14x3+274xy2

Therefore, (12x+32y)3+(12x32y)3=14x3+274xy2


3. (2 – 3x)3 – (5 + 3x)3

Solution:

(2 – 3x)3 – (5 + 3x)3

= {23 - 3.22.(3x) + 3.2.(3x)2 - (3x)3} – {53 + 3.52.(3x) + 3.5.(3x)2 + (3x)3}

= {8 – 36x + 54 x2 - 27 x3} – {125 + 225x + 135x2 + 27 x3}

= 8 – 36x + 54 x2 - 27 x3 – 125 - 225x - 135x2 - 27 x3

= 8 – 125 – 36x - 225x + 54 x2 - 135x2 - 27 x3 - 27 x3

= -117 – 261x - 81 x2 - 54 x3

Therefore, (2 – 3x)3 – (5 + 3x)3 = -117 – 261x - 81 x2 - 54 x3


4. (5m + 2n)3 - (5m – 2n)3

Solution:

(5m + 2n)3 - (5m – 2n)3

= {(5m)3 + 3.(5m)2. (2n) + 3. (5m). (2n)2 + (2n)3} – {(5m)3 - 3.(5m)2. (2n) + 3. (5m). (2n)2 - (2n)3}

= {125 m3 + 150 m2 n + 60 m n2 + 8 n3} – {125 m3 - 150 m2 n + 60 m n2 - 8 n3}

= 125 m3 + 150 m2 n + 60 m n2 + 8 n3 – 125 m3 + 150 m2 n - 60 m n2 + 8 n3

= 125 m3 – 125 m3 + 150 m2 n + 150 m2 n + 60 m n2 - 60 m n2 + 8 n3 + 8 n3

= 300 m2 n + 16 n3

Therefore, (5m + 2n)3 - (5m – 2n)3 = 300 m2 n + 16 n3


The steps to find the mixed problem on cube of a binomial will help us to expand the sum or difference of two cubes.







7th Grade Math Problems

8th Grade Math Practice

From Cube of a Binomial to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Worksheet on Average | Word Problem on Average | Questions on Average

    May 17, 25 05:37 PM

    In worksheet on average interest we will solve 10 different types of question. Find the average of first 10 prime numbers. The average height of a family of five is 150 cm. If the heights of 4 family

    Read More

  2. How to Find the Average in Math? | What Does Average Mean? |Definition

    May 17, 25 04:04 PM

    Average 2
    Average means a number which is between the largest and the smallest number. Average can be calculated only for similar quantities and not for dissimilar quantities.

    Read More

  3. Problems Based on Average | Word Problems |Calculating Arithmetic Mean

    May 17, 25 03:47 PM

    Here we will learn to solve the three important types of word problems based on average. The questions are mainly based on average or mean, weighted average and average speed.

    Read More

  4. Rounding Decimals | How to Round a Decimal? | Rounding off Decimal

    May 16, 25 11:13 AM

    Round off to Nearest One
    Rounding decimals are frequently used in our daily life mainly for calculating the cost of the items. In mathematics rounding off decimal is a technique used to estimate or to find the approximate

    Read More

  5. Worksheet on Rounding Off Number | Rounding off Number | Nearest 10

    May 15, 25 05:12 PM

    In worksheet on rounding off number we will solve 10 different types of problems. 1. Round off to nearest 10 each of the following numbers: (a) 14 (b) 57 (c) 61 (d) 819 (e) 7729 2. Round off to

    Read More