Assume that a and b are two variables that, in algebraic form, stand in for two terms. In mathematics, the sum of the two terms is denoted by the symbol(a+b). It is both a binomial and an algebraic expression. In mathematics, the cube of the sum of the terms a and b, or a binomial, is represented by the following notation:(a+b)3
This Story also Contains
- Introduction of a+b whole cube
- Usage
- Examples
- Proofs
Introduction of a+b whole cube
When a cube is added with b cube and then the result is added with three times a multiply by b, and addition of a and b, then the a plus b whole cube is obtained.
This can be represented as-
(a + b)3 =a³ + b3 + 3ab(a+b)
The formula for a plus b whole cube can also be written as a cube plus b cube plus three a square b plus three a, b square. It is written below-
(a + b)3 = a3+ b3 + 3a2b+3ab2
In mathematics, the plus b whole cubed algebraic identity is called in the following three ways-
The cube of the sum of two terms rules.
The cube of a binomial identity.
The special binomial product formula.
A:The a+b whole cube formula, also known as the cube of a binomial, is (a+b)³ = a³ + 3a²b + 3ab² + b³. This formula expands the cube of the sum of two terms, showing all the terms that result from this operation.
A:The a+b whole cube formula is an important algebraic identity. It's part of a family of identities including the square of a binomial (a+b)² and the difference of cubes a³-b³. These identities are fundamental tools in algebra for expanding and factoring expressions.
A:The a+b whole cube formula expands (a+b)³, while the square of a binomial expands (a+b)². The cube formula has four terms: a³, 3a²b, 3ab², and b³, whereas the square formula has three terms: a², 2ab, and b².
A:(a+b)³ expands to a³ + 3a²b + 3ab² + b³, while a³+b³ is just the sum of the cubes of a and b. The difference between these expressions, 3a²b + 3ab², is called the "middle terms" of the cube expansion.
A:The a+b whole cube formula is a specific case of the binomial theorem where n=3. The binomial theorem provides a general formula for expanding (a+b)ⁿ for any positive integer n.
Usage
In the following two instances, the cube of the sum of two terms rule is applied as a formula.
1. Expansion
The total of the cubes of the two terms and their product, multiplied by three, form the cube of the sum of the two terms.
(a + b)3 = a3+ b3 + 3ab(a+b)
2.Simplification
The cube of the sum of two terms is the total of the cubes of the two terms plus three times the product of the two terms added together. a3+ b3 + 3ab(a+b)=(a + b)3
A:The a+b whole cube formula can be visualized as the volume expansion of a cube with side length (a+b). Each term represents a part of the expanded volume: a³ is a large cube, 3a²b and 3ab² are rectangular prisms, and b³ is a small cube.
A:The a+b whole cube formula can be useful in polynomial long division when the dividend is a perfect cube. Recognizing the expanded form can help in identifying factors and simplifying the division process.
A:No, the a+b whole cube formula is specifically for binomials (two terms). For expressions with more than two terms, you would need to use more complex multinomial expansions or the general binomial theorem.
A:While there are no explicit squared terms, a² and b² are present within the formula. They appear in the middle terms 3a²b and 3ab². The absence of pure squared terms reflects the fact that we're dealing with a cube, not a square.
A:Yes, the formula works for complex numbers as well. You would apply it in the same way, but you need to be careful with the arithmetic of complex numbers when expanding each term.
Examples
Example 1 - Solve (2 + 3)3
Answer- We know the formula a3+ b3 + 3ab(a+b)=(a + b)3
So here a = 2 and b = 3
So we will put these values in the formula, after putting the values we will get-
23+ 33 + 3*2*3*(2+3)=(2 + 3)3
= 8+27+18 *5
=8+27+90
=125
Hence, 125 is the answer.
Example 2 - Solve (1e + 2d)3
Answer- We know the formula a3+ b3 + 3ab(a+b)=(a + b)3
So here a = 1e and b = 2d
So we will put these values in the formula, after putting the values we will get-
(1e)3+ (2d)3 + 3*1e*2d*(1e+2d)=(1e + 2d)3
= e3+8d3+6ed *(1e+2d)
=e3+8d3+6e2d+12ed2
Hence, e3+8d3+6e2d+12ed2 is the answer.
Example 3 - Solve (3h + i)3
Answer- We know the formula a3+ b3 + 3ab(a+b)=(a + b)3
So here a = 3h and b = i
So we will put these values in the formula, after putting the values we will get-
(3h)3+ i3 + 3*3h*i*(3h+i)=(3h + i)3
= 27h3+i3+9hi *(3h+i)
=27h3+i3+27h2i+9hi2
Hence , 27h3+i3+27h2i+9hi2 is the answer.
A:The a+b whole cube formula expands a perfect cube trinomial. If you can factor an expression into the form of this formula, then the original expression is a perfect cube and can be written as (a+b)³.
A:Yes, the formula can be used in reverse for factoring. If you encounter an expression of the form a³ + 3a²b + 3ab² + b³, you can factor it as (a+b)³.
A:Geometrically, (a+b)³ represents the volume of a cube with side length (a+b). The individual terms represent smaller volumes: a³ (a large cube), 3a²b (three rectangular prisms), 3ab² (three more rectangular prisms), and b³ (a small cube).
A:The formula remains the same, but you need to be careful with signs. For example, (-a+b)³ = (-a)³ + 3(-a)²b + 3(-a)b² + b³ = -a³ + 3a²b - 3ab² + b³.
A:You can check your work by expanding (a+b)(a+b)(a+b) manually and comparing the result to your application of the formula. Alternatively, you can substitute numerical values for a and b and verify that both sides of the equation are equal.
Proofs
The following two distinct mathematical techniques can be used to demonstrate the algebraic identity of a plus b for the full cube.
1. Algebraic approach
It is beneficial to use the product of three sum basis binomials to get the expansion of the a plus b whole cube formula.
2. Geometric approach
A cube's volume can be used to graphically demonstrate how the algebraic identity for a plus b entire cube can be expanded.
A:The four terms in the a+b whole cube formula represent all possible combinations of a and b when multiplied three times. Each term corresponds to choosing a or b in each of the three factors: (a+b)(a+b)(a+b).
A:The coefficients (1, 3, 3, 1) in the formula represent the number of ways to choose a or b in each term. For example, there are 3 ways to choose two a's and one b, hence the coefficient 3 in the term 3a²b.
A:You can derive the formula by multiplying (a+b) by itself three times: (a+b)(a+b)(a+b). Alternatively, you can use the binomial theorem with n=3, which gives the same result.
A:The coefficients in the a+b whole cube formula (1, 3, 3, 1) correspond to the fourth row of Pascal's triangle. This relationship holds for higher powers as well, with each row representing coefficients for successive powers.
A:If you replace b with -b, you get the a-b whole cube formula: (a-b)³ = a³ - 3a²b + 3ab² - b³. Notice that the signs of the terms alternate, but the coefficients remain the same.
