Problem 32 - Algebra and Trigonometry

Question

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ (x+3)^{8} $$

Step-by-Step Solution

Answer

The first three terms of the binomial expansion \((x+3)^8\) are \(x^8, 24x^7, 252x^6\)

Step 1: Identify the values of a, b, and n

For the binomial \((x+3)^8\), \(a=x\), \(b=3\), and \(n=8\)

Step 2: Calculate the first term of the binomial expansion

Using the Binomial Theorem, the first term is given by \(C(n, 0)a^n b^0\) which yields \(C(8, 0)x^8 * 3^0\). From the definition of the binomial coefficient, \(C(8, 0)\) equals 1, and any number to the power 0 equals 1. Thus, the first term simplifies to \(x^8\)

Step 3: Calculate the second term of the binomial expansion

The second term is given by \(C(n, 1)a^{n-1}b\), which yields \(C(8, 1)x^7 * 3^1\). Using the definition of the binomial coefficient, \(C(8, 1) = 8\), thus, the second term simplifies to \(24x^7\)

Step 4: Calculate the third term of the binomial expansion

Finally, the third term of the binomial expansion is calculated similarly to the first two terms: \(C(n, 2)a^{n-2}b^2\), which yields \(C(8, 2)x^6 * 3^2\). Using the definition of the binomial coefficient, \(C(8, 2)=28\), and simplifying, this gives \(28x^6 * 9 = 252x^6\)