Question

factor completely (3x^4 - 30x^3 + 75x^2).

• (3(x - 5)^2)
• (3x^2(x - 5)^2)
• (3x^2(x^2 - 10x + 25))
• (3x^2(x + 5)(x - 5))

question 8 (multiple choice worth 1 points)
(02.01r)

one of the factors of (p^5 - 144p^3) is

• (p^3)
• (p + 12)
• (p - 12)
• all of the above

Explanation:

Factor out the greatest common factor of the first polynomial

Using the Greatest Common Factor and Factoring Polynomials Completely knowledge points

$$ 3x^4 - 30x^3 + 75x^2 = 3x^2(x^2 - 10x + 25) $$

Factor the remaining quadratic trinomial

Using the Factoring Trinomials knowledge point

$$ x^2 - 10x + 25 = (x - 5)^2 $$

$$ 3x^4 - 30x^3 + 75x^2 = 3x^2(x - 5)^2 $$

Factor out the greatest common factor of the second polynomial

Using the Greatest Common Factor and Factoring Polynomials Completely knowledge points

$$ p^5 - 144p^3 = p^3(p^2 - 144) $$

Factor the remaining difference of squares

Using the Difference of Squares knowledge point

$$ p^2 - 144 = (p - 12)(p + 12) $$

$$ p^5 - 144p^3 = p^3(p - 12)(p + 12) $$

Identify the correct factors from the options

Using the Factoring Polynomials Completely knowledge point

$$ \text{Factors of } p^5 - 144p^3 \text{ include: } p^3, \quad p - 12, \quad p + 12 $$

Since all listed options are factors, the correct choice is "All of the above".

Answer:

Question 1

• (A) \(3(x - 5)^2\)
• (B) \(3x^2(x - 5)^2\) (Correct answer)
• (C) \(3x^2(x^2 - 10x + 25)\)
• (D) \(3x^2(x + 5)(x - 5)\)

Question 2

• (A) \(p^3\)
• (B) \(p + 12\)
• (C) \(p - 12\)
• (D) All of the above (Correct answer)