Question

Find the coefficient of x and constant term in the product (x + 3) (x + 4) (x + 5).

Answer 1

Given: Find the coefficient of x and the constant term in the product (x + 3)(x + 4)(x + 5).

Step-wise calculation:

1. Use the identity for the product of three linear factors in x:

(x + a)(x + b)(x + c) = x³ + (a + b + c)x² + (ab + bc + ca)x + abc

2. For the given expression (x + 3)(x + 4)(x + 5):

• (a = 3), (b = 4), (c = 5)

3. Calculate the sum (a + b + c):

3 + 4 + 5 = 12

4. Calculate the sum of products (ab + bc + ca):

(3 × 4) + (4 × 5) + (5 × 3) = 12 + 20 + 15

(3 × 4) + (4 × 5) + (5 × 3) = 47

5. Calculate the product (abc):

3 × 4 × 5 = 60

6. So, expanding:

(x + 3)(x + 4)(x + 5) = x³ + 12x² + 47x + 60

Coefficient of (x) = 47

Constant term = 60