POS expression
(A + B + C)(A + B + D)(A + B + C + D)
The given expression is an algebraic expression in terms of variables A, B, C, and D, which represents the product of three expressions. The expression can be simplified as follows:
(A + B + C)(A + B + D)(A + B + C + D)
Expanding the first two terms:
(A² + AB + AC + AB + B² + BC)(A + B + C + D)
Combining like terms:
(A² + 2AB + AC + B² + BC)(A + B + C + D)
Expanding the last term:
(A² + 2AB + AC + B² + BC)(A + B + C) + (A² + 2AB + AC + B² + BC)(D)
Simplifying further:
A³ + 2A²B + A²C + AB² + ABC + A²C + 2ABC + AC² + AB² + ABC + B³ + B²C + BC² + A² + 2AB + AC + B² + BC + AD
After combining like terms and simplifying, the final expression will vary depending on the values assigned to the variables A, B, C, and D.
To simplify the given expression, we can use the distributive property. Let's break it down step-by-step:
1. Start with the first two terms: (A + B + C)(A + B + D)
2. Apply the distributive property to expand these terms:
(A + B + C)(A + B + D) = A(A + B + D) + B(A + B + D) + C(A + B + D)
3. Multiply each term inside the parentheses:
A(A + B + D) = A^2 + AB + AD
B(A + B + D) = BA + B^2 + BD
C(A + B + D) = CA + CB + CD
4. Rewrite the expanded terms:
A(A + B + D) + B(A + B + D) + C(A + B + D) = A^2 + AB + AD + BA + B^2 + BD + CA + CB + CD
5. Combine like terms (AB and BA can be combined since addition is commutative):
A^2 + AB + AD + BA + B^2 + BD + CA + CB + CD = A^2 + 2AB + AD + B^2 + BD + CA + CB + CD
Now let's continue with the next term: (A + B + C + D)
6. Apply the distributive property again to expand this term:
(A + B + C + D) = A + B + C + D
7. Multiply each term by the expanded expressions from step 5:
(A^2 + 2AB + AD + B^2 + BD + CA + CB + CD)(A + B + C + D) = (A^2 + 2AB + AD + B^2 + BD + CA + CB + CD)(A + B + C + D)
Distribute each term in the expanded expression:
= A(A^2 + 2AB + AD + B^2 + BD + CA + CB + CD) + B(A^2 + 2AB + AD + B^2 + BD + CA + CB + CD) + C(A^2 + 2AB + AD + B^2 + BD + CA + CB + CD) + D(A^2 + 2AB + AD + B^2 + BD + CA + CB + CD)
Multiply each term inside the parentheses:
= A^3 + 2A^2B + A^2D + AB^2 + ABD + ACA + ACB + ACD + BA^2 + 2AB^2 + ABD + B^3 + B^2D + BCA + BCB + BCD + CA^2 + 2CAB + CAD + CB^2 + CBD + C^2A + C^2B + C^2D + DA^2 + 2DAB + DAD + DB^2 + DBD + DCA + DCB + DCD
8. Combine like terms, grouping them by degree:
A^3 + (2A^2B + AB^2) + (2AB^2 + B^3) + (2A^2B + 2AB^2 + CA^2 + C^2A + DA^2 + 2DAB) + (CAD + ACA) + (ACB + BCA + C^2B) + (A^2D + B^2D + CBD + DB^2) + (ACD + BCD + DCA + DCB) + (C^2D + DBD + CBD) + DAD + DCD
Simplifying further:
A^3 + 3A^2B + 3AB^2 + B^3 + 3A^2B + 2CA^2 + C^2A + DA^2 + 2DAB + 2CAD + 2ACA + ACB + BCA + C^2B + A^2D + B^2D + 2CBD + DB^2 + 2ACD + 2BCD + 2DCA + 2DCB + C^2D + 2DBD + DAD + 2DCD
This is the simplified form of the given expression:
A^3 + 4A^2B + 3AB^2 + B^3 + 2CA^2 + C^2A + DA^2 + 2DAB + 2CAD + 2ACA + ACB + BCA + C^2B + A^2D + B^2D + 2CBD + DB^2 + 2ACD + 2BCD + 2DCA + 2DCB + C^2D + 2DBD + DAD + 2DCD
The given expression is (A + B + C)(A + B + D)(A + B + C + D).
To simplify this expression, we can use the distributive property of multiplication over addition. According to this property, multiplying a sum/difference by another expression is equivalent to multiplying each term in the sum/difference separately, and then adding the results.
Let's break it down step by step:
Step 1: Multiply (A + B + C) with (A + B + D)
To simplify this part, we use the distributive property.
(A + B + C)(A + B + D) = A(A + B + D) + B(A + B + D) + C(A + B + D)
Step 2: Simplify each term in Step 1
A(A + B + D) = A^2 + AB + AD
B(A + B + D) = AB + B^2 + BD
C(A + B + D) = CA + CB + CD
Step 3: Combine the terms from Step 2
(A + B + C)(A + B + D) = A^2 + AB + AD + AB + B^2 + BD + CA + CB + CD
Step 4: Simplify further
Combine like terms:
A^2 + (AB + AB) + AD + B^2 + (BD + CD) + (CA + CB)
A^2 + 2AB + AD + B^2 + BD + CD + CA + CB
Step 5: Multiply the expression from Step 4 with (A + B + C + D)
To simplify this part, we use the distributive property once again.
(A^2 + 2AB + AD + B^2 + BD + CD + CA + CB)(A + B + C + D) = A(A^2 + 2AB + AD + B^2 + BD + CD + CA + CB) + B(A^2 + 2AB + AD + B^2 + BD + CD + CA + CB) + C(A^2 + 2AB + AD + B^2 + BD + CD + CA + CB) + D(A^2 + 2AB + AD + B^2 + BD + CD + CA + CB)
Step 6: Simplify each term in Step 5
Multiply each term separately by A, B, C, and D:
A(A^2 + 2AB + AD + B^2 + BD + CD + CA + CB) = A^3 + 2A^2B + A^2D + AB^2 + ABD + ACD + ACA + ACB
B(A^2 + 2AB + AD + B^2 + BD + CD + CA + CB) = A^2B + 2AB^2 + ABD + B^3 + B^2D + BCD + BCA + BCB
C(A^2 + 2AB + AD + B^2 + BD + CD + CA + CB) = A^2C + 2ABC + ACD + BC^2 + BCD + CD^2 + C^2A + C^2B
D(A^2 + 2AB + AD + B^2 + BD + CD + CA + CB) = A^2D + 2ABD + AD^2 + B^2D + BD^2 + CDD + CAD + CDB
Step 7: Combine the terms from Step 6
Add all the terms together:
A^3 + 2A^2B + A^2D + AB^2 + ABD + ACD + ACA + ACB + A^2B + 2AB^2 + ABD + B^3 + B^2D + BCD + BCA + BCB + A^2C + 2ABC + ACD + BC^2 + BCD + CD^2 + C^2A + C^2B + A^2D + 2ABD + AD^2 + B^2D + BD^2 + CDD + CAD + CDB
Step 8: Simplify further
Combine like terms:
A^3 + 3A^2B + 2A^2D + AB^2 + 2ABD + 2ACD + ACA + ACB + B^3 + B^2D + 2BCD + BCA + BCB + A^2C + 2ABC + BC^2 + CD^2 + C^2A + C^2B + AD^2 + 2ABD + BD^2 + CDD + CAD + CDB
Now, the simplified expression is:
A^3 + 3A^2B + 2A^2D + AB^2 + 2ABD + 2ACD + ACA + ACB + B^3 + B^2D + 2BCD + BCA + BCB + A^2C + 2ABC + BC^2 + CD^2 + C^2A + C^2B + AD^2 + 2ABD + BD^2 + CDD + CAD + CDB