Calculate the term independent of x in the expansion of (2x^2 - 3x + 4)^3.
Practice Questions
1 question
Q1
Calculate the term independent of x in the expansion of (2x^2 - 3x + 4)^3.
12
24
36
48
The term independent of x occurs when the powers of x cancel out. The term is C(3,0)(4)^3 + C(3,1)(-3x)(4)^2 + C(3,2)(-3x)^2(4) + C(3,3)(-3x)^3 = 64 - 36 + 0 + 0 = 28.
Questions & Step-by-step Solutions
1 item
Q
Q: Calculate the term independent of x in the expansion of (2x^2 - 3x + 4)^3.
Solution: The term independent of x occurs when the powers of x cancel out. The term is C(3,0)(4)^3 + C(3,1)(-3x)(4)^2 + C(3,2)(-3x)^2(4) + C(3,3)(-3x)^3 = 64 - 36 + 0 + 0 = 28.
Steps: 16
Step 1: Identify the expression to expand, which is (2x^2 - 3x + 4)^3.
Step 2: Understand that we need to find the term that does not contain 'x' (the term independent of x).
Step 3: Use the binomial theorem to expand the expression. We will look for combinations of terms that will cancel out the 'x' powers.
Step 4: Write down the general term in the expansion: C(3, k) * (2x^2)^(3-k) * (-3x)^k * 4^(3-k), where k is the number of times we choose the -3x term.
Step 5: Calculate the terms for k = 0, 1, 2, and 3 to find the independent term.
Step 10: Now, we need to find the constant term. We can also calculate the constant term directly: C(3,0)(4)^3 + C(3,1)(-3x)(4)^2 + C(3,2)(-3x)^2(4) + C(3,3)(-3x)^3.
