Question

factor \\(x^4 - 81\\).

a) \\((x^2 + 3)(x^2 - 3)\\)
b) \\((x^2 + 9)(x + 3)(x - 3)\\)
c) \\((x + 3)^2(x - 3)^2\\)
d) \\((x^2 - 9)(x + 3)(x - 3)\\)

question 12 (5 points)
consider points \\(a\\), \\(b\\), and \\(c\\) in the graph. determine which of these points is relative minima on the interval \\(x = -1\\) and \\(x = -2\\) in the graph.

a) \\(a\\) and \\(b\\)
b) \\(c\\)
c) \\(b\\)
d) \\(a\\)

Explanation:

Factor the expression

To factor the expression \(x^4 - 81\), we recognize it as a difference of squares:

$$ x^4 - 81 = (x^2)^2 - 9^2 $$

Applying the difference of squares formula \(a^2 - b^2 = (a - b)(a + b)\):

$$ x^4 - 81 = (x^2 + 9)(x^2 - 9) $$

The term \(x^2 - 9\) is also a difference of squares, \(x^2 - 3^2\), which factors further:

$$ x^2 - 9 = (x + 3)(x - 3) $$

Substituting this back gives the fully factored form:

$$ x^4 - 81 = (x^2 + 9)(x + 3)(x - 3) $$

This matches option B.

Identify the relative minimum

For Question 12, we need to find which point is a relative minimum on the interval between \(x = -1\) and \(x = -2\) (noting the prompt's typo "interval \(x = -1\) and \(x = -2\)", which refers to the interval \([-2, -1]\)).
Looking at the graph:

• Point \(a\) is located at approximately \((-1.5, -2.5)\), which is the lowest point in its local neighborhood.
• Point \(b\) is on a downward slope near \(x = 2\).
• Point \(c\) is on a downward slope near \(x = -3\).

A relative minimum is a point where the function transitions from decreasing to increasing. Point \(a\) is the local valley in this region. Thus, point \(a\) is the relative minimum on the interval \([-2, -1]\). This matches option D.

Answer:

Question 1

• (A) \((x^2 + 3)(x^2 - 3)\)
• (B) \((x^2 + 9)(x + 3)(x - 3)\) (Correct answer)
• (C) \((x + 3)^2(x - 3)^2\)
• (D) \((x^2 - 9)(x + 3)(x - 3)\)

Question 12

• (A) \(a\) and \(b\)
• (B) \(c\)
• (C) \(b\)
• (D) \(a\) (Correct answer)