Verified answer

1)

x^2 + 4x + 4

= x^2 + 2x + 2x + 4

= (x^2 + 2x) + (2x + 4)

= x(x + 2) + 2(x + 2)

= (x + 2)(x + 2)

= (x + 2)^2

2)

25x^2 + 10x + 1

= 25x^2 + 5x + 5x + 1

= (25x^2 + 5x) + (5x + 1)

= 5x(5x + 1) + 1(5x + 1)

= (5x + 1)(5x + 1)

= (5x + 1)^2

3)

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

x^3 - 27 = x^3 - 3^3 = (x - 3)(x^2 + 3x + 9)

I'll try to explain how to factor the two quadratic equations. I'll use a formula for the cubic equation.

I'll use * for the multiplication symbol since we are already using x.

1) Since the coefficient of x^2 is 1, we are looking for two numbers, a and b such that a+b = 4 (for the middle term) and a*b = 4 for the constant term. The answer will be (x+a)*(x+b)

4 = 2*2 and 1*4

2+2 = 4, but 1 + 4 = 5 so a = b = 2

Answer (x+2)*(x+2)

2) Since the coefficient of the constant term is 1, we are looking for two numbers a and b such that a+b = 10 (for the middle term) and a*b=25 for the coefficient of the x^2 term. The answer will be (ax+1)*bx+1).

25 = 5*5 and 1*25

5+5=10 but 1+25=26 so a = b = 5

Answer: (5x+1)*(5x+1)

3) For the difference of two perfect cubes

a^3 - b^3 = (a-b)(a^2+ab+b^2)

Therefore: x^3 - 27 = (x - 3)*(x^2 + 3x + 9)

Not many options on this one. It's a perfect square.

x² + 4x + 4 = 0

(x + 2)² = 0

Pretty easy to see this one too. Also a perfect square.

25x² + 10x + 1 = 0

(5x + 1)² = 0

The last one is the difference of two cubes, so use the formula (a³ - b³) = (a - b)(a² + ab + b²)

x³ - 27 = 0

(x - 3)(x² + 3x + 9) = 0

Factor completely, if possible

x^2 + 4x + 4 = (x + 2)(x + 2)

Factor completely, if possible

25x^2 + 10x + 1 = (5x + 1)(5x + 1)

Factor completely, if possible

x^3 -- 27 = (x -- 3)(x^2 + 3x + 9)

I can't do those exact problems for you, but I can give you an example problem to help you figure this out.

x^3 - 6 + 2x - 3x^2

The terms x^3 and -6 have no common factor. Use the commutative property to rearrange the terms so that you can group terms with a common factor.

x^3 - 3x^2 + 2x -6

So basically put the ones with exponents on one side, and ones without exponents on the other.

Now group the terms

(x^3 - 3x^2) + (2x-6)

Now Factor

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

Finally, use the distributive property.

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

DONE!!!

3x² + 15x - forty two is the equation. yet wait, the coefficients are all divisible by skill of three. (3, 15, and forty two) So place a three in front of the completed element, and divide by skill of three interior the parenthesis. you should get this...... 3 (x² + 5x - 14) now we ought to ingredient the trinomial interior the parenthesis, it fairly is (x² + 5x - 14). the effect ought to look something like this: 3 (x ____ ) (x ____) think of, what 2 numbers prolonged supply you 14 (0.33 term), and 5 (center term) while they are further/subtracted. in case you probably did it suitable, you will have considered 7 and a couple of, suitable? 7 x 2 = 14 ???????? 7 - 2 = 5 Now place the 7 and a couple of with the superb signs and warning signs interior the blanks. 3 (x + 7) (x - 2) and that's your answer!

(x+2)(x+2)

(5x+1)(5x+1)