Question

Factor completely 3x4 − 30x3 + 75x2. 3(x − 5)2 3x2(x − 5)2 3x2(x2 − 10x + 25) 3x2(x + 5)(x − 5)

Answer 1

3x^2 (x-5)^2

Explanation:

3x^4 − 30x^3 + 75x^2

We can factor out a 3x^2 from each term

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

The term inside the parentheses can be factored

What 2 numbers multiply to 25 and add to -10

-5*-5 = 25

-5+-5 = -10

3x^2 (x-5) (x-5)

3x^2 (x-5)^2

Answer 2

The complete factorization is 3x² (x - 5)² ⇒ 2nd answer

Explanation:

* Lets revise how to factorize a trinomial

- Find the greatest common factor of the coefficients of the three terms

∵ The trinomial is 3x^4 - 30x³ + 75x²

- The greatest common factor of 3 , 30 , 75 is 3

∵ 3 ÷ 3 = 1

∵ 30 ÷ 3 = 10

∵ 75 ÷ 3 = 25

∴ 3x^4 - 30x³ + 75x² = 3(x^4 - 10x³ + 25x²)

- Now lets find the greatest common factor of the variable x

∵ x² is the greatest common factor of the three terms

∵ x^4 ÷ x² = x²

∵ 10x³ ÷ x² = 10x

∵ 25x² ÷ x² = 25

∴ 3(x^4 - 10x³ + 25x²) = 3x² (x² - 10x + 25)

- Lets factorize (x² - 10x + 25)

∵ √x² = x

∵ √25 = 5

∵ 2 × 5 × x = 10x

∴ x² - 10x + 25 is a completing square

∴ (x² - 10x + 25) = (x - 5)²

∴ 3x² (x² - 10x + 25) = 3x² (x - 5)²

* The complete factorization is 3x² (x - 5)²