Using Pascal's Triangle, what is the third term in the expansion of (a+5)^5 when expanded polynomial is written in standard form? (1 point)
a. 10a^3
b. 250a^3
c. 250a^2
d. 25a^3
To find the third term in the expansion of (a+5)^5, we use Pascal's Triangle. Pascal's Triangle gives us the coefficients for each term in the expansion.
The coefficients for the expansion of (a+5)^5 are the numbers in the fifth row of Pascal's Triangle: 1 5 10 10 5 1.
The third term in the expansion corresponds to the third number, which is 10.
Therefore, the third term in the expansion is 10a^3.
The correct answer is a. 10a^3.
To find the third term in the expansion of (a+5)^5, we can use Pascal's Triangle.
Pascal's Triangle is a triangular array of numbers in which each number is the sum of the two numbers directly above it.
To find the third term, we need to find the coefficients of the terms in the expansion. The third term will have an exponent of a^3.
The coefficients in the expansion of (a+5)^5 are given by the fifth row of Pascal's Triangle: 1, 5, 10, 10, 5, 1.
Therefore, the coefficient of the third term in the expansion of (a+5)^5 is 10.
The third term in the expansion of (a+5)^5 when written in standard form is 10a^3.
Therefore, the correct answer is a. 10a^3.
To find the third term in the expansion of (a+5)^5, we can use Pascal's Triangle. Pascal's Triangle is a triangular array of numbers in which each number is the sum of the two numbers directly above it.
To expand (a+5)^5, we need to raise each term (a and 5) separately to the powers from 0 to 5 and then multiply them with the corresponding coefficients from Pascal's Triangle.
The coefficients of the expansion (a+5)^5 are given by the row with 5 as its index in Pascal's Triangle, which is 1 5 10 10 5 1.
The terms in the expansion correspond to these coefficients multiplied by the respective powers of a and 5. Since we want to find the third term, we need to look at the term with a cubed (a^3) power.
The third term in the expansion is obtained by taking the third coefficient (10) and multiplying it by a^3 and 5^2 since the powers a^3 and 5^2 add up to 5.
So, the third term is 10 * a^3 * 5^2 = 10 * a^3 * 25 = 250a^3.
Therefore, the correct answer is option b. 250a^3.