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A plus B Cube Formula: Definition and Examples | EDU.COM

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A plus B Cube Formula: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">A plus B Cube FormulaA plus B Cube Formula: Definition and ExamplesTable of ContentsThe A plus B Cube Formula: Cube of a Binomial Definition of the A plus B Cube Formula

The (a+b)3(a+b)^3(a+b)3 formula represents the cube of a binomial, which helps us simplify algebraic expressions. This formula expands to (a+b)3=a3+3a2b+3ab2+b3(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3(a+b)3=a3+3a2b+3ab2+b3, showing how the cube of the sum of two terms can be broken down into four distinct terms. The formula works for any values of 'aaa' and 'bbb', making it a versatile tool in algebra.

The coefficients in the expansion (1,3,3,1)(1, 3, 3, 1)(1,3,3,1) match the fourth row of Pascal's Triangle, showing the connection between binomial expansions and this mathematical pattern. The formula can also be expressed as (a+b)3=(a+b)×(a+b)×(a+b)(a+b)^3 = (a+b) \times (a+b) \times (a+b)(a+b)3=(a+b)×(a+b)×(a+b), or alternatively as (a+b)3=a3+3ab(a+b)+b3(a+b)^3 = a^3 + 3ab(a+b) + b^3(a+b)3=a3+3ab(a+b)+b3. By the commutative property of addition, we also know that (a+b)3=(b+a)3(a+b)^3 = (b+a)^3(a+b)3=(b+a)3.

Examples of the A plus B Cube Formula Example 1: Expanding a Binomial with a Variable and a Constant Problem:

Expand (x+2)3(x + 2)^3(x+2)3.

Step-by-step solution:

Step 1, Write out the (a+b)3(a+b)^3(a+b)3 formula. We know that: (a+b)3=a3+3a2b+3ab2+b3(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3(a+b)3=a3+3a2b+3ab2+b3

Step 2, Identify the values of a and b in our problem. Here, a=xa = xa=x and b=2b = 2b=2

Step 3, Substitute these values into the formula:

(x+2)3=x3+3x2(2)+3x(2)2+23(x+2)^3 = x^3 + 3x^2(2) + 3x(2)^2 + 2^3(x+2)3=x3+3x2(2)+3x(2)2+23

Step 4, Calculate each term:

(x+2)3=x3+3x2(2)+3x(2)2+23(x+2)^3 = x^3 + 3x^2(2) + 3x(2)^2 + 2^3(x+2)3=x3+3x2(2)+3x(2)2+23

(x+2)3=x3+6x2+12x+8(x+2)^3 = x^3 + 6x^2 + 12x + 8(x+2)3=x3+6x2+12x+8

Example 2: Using the Formula to Evaluate a Numerical Expression Problem:

Evaluate 14314^3143 using the (a+b)3(a+b)^3(a+b)3 formula.

Step-by-step solution:

Step 1, Break down 141414 into a sum that makes calculation easier. We can write 14=10+414 = 10 + 414=10+4

Step 2, Use the $(a+b)^3$ formula with a=10a = 10a=10 and b=4b = 4b=4:

(a+b)3=a3+3a2b+3ab2+b3(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3(a+b)3=a3+3a2b+3ab2+b3

(10+4)3=103+3(10)2(4)+3(10)(4)2+43(10+4)^3 = 10^3 + 3(10)^2(4) + 3(10)(4)^2 + 4^3(10+4)3=103+3(10)2(4)+3(10)(4)2+43

Step 3, Calculate each term:

(10+4)3=1000+3(100)(4)+3(10)(16)+64(10+4)^3 = 1000 + 3(100)(4) + 3(10)(16) + 64(10+4)3=1000+3(100)(4)+3(10)(16)+64

(10+4)3=1000+1200+480+64(10+4)^3 = 1000 + 1200 + 480 + 64(10+4)3=1000+1200+480+64

Step 4, Add all terms to get the final answer:

(10+4)3=2744(10+4)^3 = 2744(10+4)3=2744

143=274414^3 = 2744143=2744

Example 3: Finding the Coefficient in a Binomial Expansion Problem:

Find the coefficient of the term a2ba^2ba2b in the expansion of (3a+2b)3(3a + 2b)^3(3a+2b)3.

Step-by-step solution:

Step 1, Use the (a+b)3(a+b)^3(a+b)3 formula with a=3aa = 3aa=3a and b=2bb = 2bb=2b:

(a+b)3=a3+3a2b+3ab2+b3(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3(a+b)3=a3+3a2b+3ab2+b3

(3a+2b)3=(3a)3+3(3a)2(2b)+3(3a)(2b)2+(2b)3(3a+2b)^3 = (3a)^3 + 3(3a)^2(2b) + 3(3a)(2b)^2 + (2b)^3(3a+2b)3=(3a)3+3(3a)2(2b)+3(3a)(2b)2+(2b)3

Step 2, Expand each term:

(3a+2b)3=27a3+3(9a2)(2b)+3(3a)(4b2)+8b3(3a+2b)^3 = 27a^3 + 3(9a^2)(2b) + 3(3a)(4b^2) + 8b^3(3a+2b)3=27a3+3(9a2)(2b)+3(3a)(4b2)+8b3

(3a+2b)3=27a3+54a2b+36ab2+8b3(3a+2b)^3 = 27a^3 + 54a^2b + 36ab^2 + 8b^3(3a+2b)3=27a3+54a2b+36ab2+8b3

Step 3, Look at the term containing a2ba^2ba2b. From our expansion, we can see it is 54a2b54a^2b54a2b.

Step 4, So the coefficient of the term a2ba^2ba2b is 545454.

Comments(1)MCMs. CarterSeptember 17, 2025I’ve been helping my daughter with algebra, and this page explained the (a+b)³ formula so clearly! The step-by-step examples made it easy for her to follow. Great resource for parents!

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