Cube of the Difference of Two Binomials

What is the formula for the cube of the difference of two binomials?

To determine cube of a number means multiplying a number with itself three times similarly, cube of a binomial means multiplying a binomial with itself three times.



(a - b) (a - b) (a - b) = (a - b)3

or, (a - b) (a - b) (a - b) = (a - b) (a - b)2

                                   = (a – b) (a2 + b2 - 2ab),
                                   [Using the formula of (a + b) 2 = a2 - 2ab + b2]

                                   = a (a2 + b2 – 2ab) – b (a2 + b2 – 2ab)

                                   = a3 + ab2 – 2a2b – ba2 – b3 + 2ab2

                                   = a3 – 3a2b + 3ab2 – b3



Therefore, (a - b)3 = a3 – 3a2b + 3ab2 – b3

Thus, we can write it as; a = first term, b = second term

(First term – Second term)3 = (first term)3 - 3 (first term)2 (second term) + 3 (first term) (second term)2 - (second term)3

So, the formula for the cube of the difference of two terms is written as:

(a - b)3 = a3 – 3a2b + 3ab2 – b3

            = a3 – b3 – 3ab (a - b)



Worked-out examples to find the cube of the difference of two binomials:

1. Determine the expansion of (3x – 4y)3

Solution:

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

(3x – 4y)3

Here, a = 3x, b = 4y

= (3x)3 – 3 (3x)2 (4y) + 3 (3x) (4y)2 – (4y)3

= 27x3 – 3 (9x2) (4y) + 3 (3x) (16y2) – 64y3

= 27x3 – 108x2y + 144xy2 – 64y3

Therefore, (3x – 4y)3 = 27x3 – 108x2y + 144xy2 – 64y3


2. Use the formula and evaluate (997)3

Solution:

(997)3 = (1000 – 3)3

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

Here, a = 1000, b = 3

(1000 – 3)3

= (1000)3 – 3 (1000)2 (3) + 3 (1000) (3)2 – (3)3

= 1000000000 – 9 (1000000) + (3000) 9 – 27

= 1000000000 – 9000000 + 27000 – 27

= 991026973

Therefore, (997)3 = 991026973

Thus, to expand the cube of the difference of two binomials we can use the formula to evaluate.









7th Grade Math Problems

8th Grade Math Practice

From Cube of The Difference of Two Binomials 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 Addition of 4-Digit Numbers | 4 Digit Addition Worksheets

    Jan 10, 25 03:02 PM

    Worksheet on Addition of 4-Digit Numbers
    Practice the questions given in the worksheet on addition of 4-digit numbers. Here we will add two 4-digit numbers (without carrying and with carrying) and three 4-digit numbers

    Read More

  2. Word Problems on 4-Digit Numbers |Addition and Subtraction of 4-Digits

    Jan 10, 25 02:49 PM

    Word Problems on 4-Digit Numbers
    We will solve here some of the word problems on addition and subtraction of 4-digit numbers. We will apply the same method while adding and subtracting the word problems. 1. In a village, there are 25…

    Read More

  3. Addition of 10, 100 and 1000 | Adding 10 | Adding 100 | Adding 1000

    Jan 10, 25 01:20 AM

    Adding 10
    Here we will learn Addition of 10, 100 and 1000 with the help of different examples.

    Read More

  4. Estimating a Sum | Round the Number | Numbers by Rounding | Estimating

    Jan 10, 25 12:10 AM

    Estimating the Sum
    We will learn the basic knowledge for estimating a sum. Here we will learn an easy way to estimate a sum of two numbers by rounding. In case of two digit numbers we can only round the number

    Read More

  5. Addition of 4-Digit Numbers | 4-Digit Addition |Adding 4-Digit Numbers

    Jan 09, 25 10:07 PM

    Addition of 4-Digit Numbers
    We will learn about the addition of 4-digit numbers (without carrying and with carrying). We know how to add 2 or 3, 3-digit numbers without carrying or with carrying.

    Read More