Is this true or false? To expand the binomial (x2 + 3)6, use the row of Pascal's triangle that has coefficients 1, 5, 10, 10, 5, 1.
This statement is true. The coefficients in the expansion of the binomial (x^2 + 3)^6 can be found using the row of Pascal's triangle that has coefficients 1, 5, 10, 10, 5, 1.
To explain how Pascal's triangle is related to expanding binomials, let's take a closer look at the triangle. Pascal's triangle is a triangular arrangement of numbers where each number is the sum of the two numbers directly above it. The top row of the triangle is considered row 0, and the numbers in each row represent the coefficients in the expansion of (a + b)^n, where a and b are variables.
For example, to expand (a + b)^3, we use the coefficients from row 3 of Pascal's triangle, which are 1, 3, 3, 1. These numbers represent the coefficients of a^3, a^2b, ab^2, and b^3 in the expansion of (a + b)^3.
In our case, we want to expand (x^2 + 3)^6. To find the coefficients, we can use the row of Pascal's triangle that corresponds to the power of the binomial. In this case, the power is 6, so we look at row 6 of Pascal's triangle. The coefficients in this row are 1, 6, 15, 20, 15, 6, 1.
Therefore, to expand (x^2 + 3)^6, we can write it as:
1(x^2)^6 + 6(x^2)^5(3) + 15(x^2)^4(3^2) + 20(x^2)^3(3^3) + 15(x^2)^2(3^4) + 6(x^2)(3^5) + 1(3^6)
Simplifying this expression will give you the expanded form of (x^2 + 3)^6.