What is the formula for (a+b)³?, How many terms are there in the expansion of (a+b)³?, What is the coefficient of the a³ term in (a+b)³?, What is the coefficient of the b³ term in (a+b)³?, What is the middle term in the expansion of (a+b)³?, What is the expansion of (a+b)³?, How does the Binomial Theorem apply to (a+b)³?, What is the sum of the coefficients in the expansion of (a+b)³?, How would you verify the expansion of (a+b)³ by multiplying (a+b)(a+b)(a+b)?, How is the expansion of (a+b)³ related to Pascal’s Triangle?

Here are the answers to the questions about the expression $(a+b{)}^3$:

1. What is the formula for $(a+b{)}^3$?

   • The formula for $(a+b{)}^3$ is $a^3+3a^{2}b+3a{b}^2+b^3$.
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2. How many terms are there in the expansion of $(a+b{)}^3$?

   • There are four terms in the expansion of $(a+b{)}^3$: $a^3$, $3a^{2}b$, $3a{b}^2$, and $b^3$.
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3. What is the coefficient of the $a^3$ term in $(a+b{)}^3$?

   • The coefficient of the $a^3$ term is 1.
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4. What is the coefficient of the $b^3$ term in $(a+b{)}^3$?

   • The coefficient of the $b^3$ term is 1.
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5. What is the middle term in the expansion of $(a+b{)}^3$?

   • The middle terms are $3a^{2}b$ and $3a{b}^2$. Since they are symmetrical, together they are considered the "middle" in terms of distribution.
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6. What is the expansion of $(a+b{)}^3$?

   • The expansion of $(a+b{)}^3$ is $a^3+3a^{2}b+3a{b}^2+b^3$.
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7. How does the Binomial Theorem apply to $(a+b{)}^3$?

   • The Binomial Theorem provides the coefficients for each term in the expansion of $(a+b{)}^n$. For $(a+b{)}^3$, the coefficients are derived from the third row of Pascal's Triangle: 1, 3, 3, 1.

8. What is the sum of the coefficients in the expansion of $(a+b{)}^3$?

   • The sum of the coefficients is $1+3+3+1=8$.
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9. How would you verify the expansion of $(a+b{)}^3$ by multiplying $(a+b)(a+b)(a+b)$?

   • To verify, multiply $(a+b)$ twice to get $(a+b{)}^2=a^2+2ab+b^2$, then multiply the result by $(a+b)$ to get $a^3+3a^{2}b+3a{b}^2+b^3$.
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10. How is the expansion of $(a+b{)}^3$ related to Pascal’s Triangle?

    • The coefficients $1,3,3,1$ in the expansion of $(a+b{)}^3$ correspond to the numbers in the third row of Pascal’s Triangle.