Question

*1. what is the correct factored form of 324v² - 196?
a. (14v + 18)(14v - 18)
b. (18v + 14)(18v - 14)
c. (14v - 18)(14v - 18)
d. (18v - 14)(18v - 14)

1. what is the correct factored form of 36x² - 81a²?

a. (6x - 9a)²
b. (6x - 9a)(30x - 72a)
c. (6x + 9a)(6x - 9a)
d. prime polynomial

*3. which of the following is the correct factored form of x⁴ - 81?
a. (x² + 9)(x + 3)(x - 3)
b. (x + 25)(x - 5)(x + 5)(x - 5)
c. (x² + 5)(x² - 5)(x + 5)(x - 5)
d. prime polynomial

Explanation:

Question 1

Step1: Identify the form

The expression \(324v^2 - 196\) is a difference of squares, since \(324v^2=(18v)^2\) and \(196 = 14^2\). The formula for factoring a difference of squares is \(a^2 - b^2=(a + b)(a - b)\).

Step2: Apply the formula

Here, \(a = 18v\) and \(b = 14\). So, \(324v^2-196=(18v + 14)(18v - 14)\).

Step1: Identify the form

The expression \(36x^2-81a^2\) is a difference of squares, since \(36x^2=(6x)^2\) and \(81a^2=(9a)^2\). The formula for factoring a difference of squares is \(a^2 - b^2=(a + b)(a - b)\).

Step2: Apply the formula

Here, \(a = 6x\) and \(b = 9a\). So, \(36x^2 - 81a^2=(6x + 9a)(6x - 9a)\).

Step1: Identify the form

The expression \(x^4-81\) can be written as \((x^2)^2-9^2\), which is a difference of squares. First, factor it using the difference of squares formula \(a^2 - b^2=(a + b)(a - b)\) where \(a=x^2\) and \(b = 9\). So, \(x^4 - 81=(x^2 + 9)(x^2 - 9)\).

Step2: Factor \(x^2-9\)

The expression \(x^2 - 9\) is also a difference of squares (\(x^2-3^2\)). Using the difference of squares formula again with \(a = x\) and \(b = 3\), we get \(x^2 - 9=(x + 3)(x - 3)\).

Step3: Combine the factors

Putting it all together, \(x^4-81=(x^2 + 9)(x + 3)(x - 3)\).

Answer:

B. \((18v + 14)(18v - 14)\)

Question 2