Question

determine the product of the polynomials using the distributive property (f.o.i.l.)

1. $(2x + 1)(x + 8)$ 4. $(- x + 3)(x^2 - 1)$

use special products to determine each product

1. $(x + 4)(x - 4)$ 6. $(4x + 3)^2$

Explanation:

Problem 3: \((2x + 1)(x + 8)\)

Step 1: Apply F.O.I.L. (First, Outer, Inner, Last)

First: \(2x \cdot x = 2x^2\)
Outer: \(2x \cdot 8 = 16x\)
Inner: \(1 \cdot x = x\)
Last: \(1 \cdot 8 = 8\)

Step 2: Combine like terms

\(2x^2 + 16x + x + 8 = 2x^2 + 17x + 8\)

Step 1: Distribute each term

\(-x \cdot x^2 + (-x) \cdot (-1) + 3 \cdot x^2 + 3 \cdot (-1)\)

Step 2: Simplify each product

\(-x^3 + x + 3x^2 - 3\)

Step 3: Rearrange terms (optional, for standard form)

\(-x^3 + 3x^2 + x - 3\)

Step 1: Use difference of squares formula \((a + b)(a - b) = a^2 - b^2\)

Here, \(a = x\), \(b = 4\)

Step 2: Apply the formula

\(x^2 - 4^2 = x^2 - 16\)

Answer:

\(2x^2 + 17x + 8\)

Problem 4: \((-x + 3)(x^2 - 1)\)