Use Pascal's triangle to expand $(x+3)^4$.
A) $x^4-12 x^3+54 x^2-108 x+81$
B) $x^4+12 x^3+54 x^2+108 x+81$
C) $x^4-20 x^3+90 x^2-135 x+81$
D) $x^4+20 x^3+90 x^2+135 x+81$
Answer (2)
Use Pascal's triangle to find the coefficients for the expansion of ( x + 3 ) 4 , which are 1, 4, 6, 4, 1.
Apply the binomial theorem: ( x + 3 ) 4 = 1 ∗ x 4 ∗ 3 0 + 4 ∗ x 3 ∗ 3 1 + 6 ∗ x 2 ∗ 3 2 + 4 ∗ x 1 ∗ 3 3 + 1 ∗ x 0 ∗ 3 4 .
Simplify the expression: ( x + 3 ) 4 = x 4 + 12 x 3 + 54 x 2 + 108 x + 81 .
The correct expansion is x 4 + 12 x 3 + 54 x 2 + 108 x + 81 .
Explanation
Understanding the Problem We are asked to expand ( x + 3 ) 4 using Pascal's triangle and choose the correct expansion from the given options.
Finding the Coefficients Pascal's triangle provides the coefficients for the binomial expansion. The coefficients for the expansion of a binomial raised to the power of 4 are found in the 5th row of Pascal's triangle (remembering that the first row is row 0). The coefficients are 1, 4, 6, 4, 1.
Applying the Binomial Theorem Using the binomial theorem, the expansion of ( x + 3 ) 4 is given by: ( x + 3 ) 4 = ( 0 4 ) x 4 3 0 + ( 1 4 ) x 3 3 1 + ( 2 4 ) x 2 3 2 + ( 3 4 ) x 1 3 3 + ( 4 4 ) x 0 3 4
Substituting the Coefficients Substituting the coefficients from Pascal's triangle into the binomial expansion, we get: ( x + 3 ) 4 = 1 ∗ x 4 ∗ 1 + 4 ∗ x 3 ∗ 3 + 6 ∗ x 2 ∗ 9 + 4 ∗ x ∗ 27 + 1 ∗ 1 ∗ 81
Simplifying the Expression Simplifying the expression, we have: ( x + 3 ) 4 = x 4 + 12 x 3 + 54 x 2 + 108 x + 81
Identifying the Correct Answer Comparing the expanded form with the given options, we find that the correct answer is: x 4 + 12 x 3 + 54 x 2 + 108 x + 81 which corresponds to option B.
Examples
Pascal's triangle and binomial expansion are not just abstract math; they have real-world applications. For instance, imagine you're calculating the probability of getting a certain number of heads when flipping a coin multiple times. If you flip a coin 4 times, the expansion of ( H + T ) 4 (where H is heads and T is tails) will give you the probabilities of each outcome: H 4 + 4 H 3 T + 6 H 2 T 2 + 4 H T 3 + T 4 . The coefficients 1, 4, 6, 4, 1 tell you how many ways you can get each combination (e.g., 4 ways to get 3 heads and 1 tail). This is used in genetics to predict the traits of offspring, in finance to assess risk, and in many other fields dealing with probabilities.
To expand ( x + 3 ) 4 using Pascal's triangle, the coefficients are found in the 5th row, which are 1, 4, 6, 4, and 1. The expanded form is x 4 + 12 x 3 + 54 x 2 + 108 x + 81 , corresponding to option B. Therefore, the correct answer is option B: x 4 + 12 x 3 + 54 x 2 + 108 x + 81 .
;
