Factor:
12x^3+18x
49-4y^2
x^3+1
8x^3-125
x^3+2x^2+x
To factor these expressions, we will apply different methods based on the given expressions.
1. Factor: 12x^3 + 18x
Step 1: We can factor out the common factor, which is 6x.
12x^3 + 18x = 6x(2x^2 + 3)
So, the factored form of 12x^3 + 18x is 6x(2x^2 + 3).
2. Factor: 49 - 4y^2
This expression is in the form of the difference of squares.
Step 1: Recognize that 49 is a perfect square (7^2), and 4y^2 is also a perfect square (2y)^2.
Step 2: Apply the formula: (a^2 - b^2) = (a + b)(a - b)
49 - 4y^2 = (7)^2 - (2y)^2
= (7 + 2y)(7 - 2y)
So, the factored form of 49 - 4y^2 is (7 + 2y)(7 - 2y).
3. Factor: x^3 + 1
Step 1: This expression is a sum of cubes.
Step 2: Apply the formula: (a^3 + b^3) = (a + b)(a^2 - ab + b^2)
x^3 + 1 = (x + 1)(x^2 - x + 1)
So, the factored form of x^3 + 1 is (x + 1)(x^2 - x + 1).
4. Factor: 8x^3 - 125
Step 1: Recognize that 8 is a perfect cube (2^3), and 125 is a perfect cube (5^3).
Step 2: Apply the formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2)
8x^3 - 125 = (2x)^3 - 5^3
= (2x - 5)((2x)^2 + (2x)(5) + 5^2)
= (2x - 5)(4x^2 + 10x + 25)
So, the factored form of 8x^3 - 125 is (2x - 5)(4x^2 + 10x + 25).
5. Factor: x^3 + 2x^2 + x
Step 1: This expression is a trinomial.
Step 2: Look for common factors, if any. In this case, there is a common factor of x.
x^3 + 2x^2 + x = x(x^2 + 2x + 1)
Step 3: Factor the remaining quadratic expression, x^2 + 2x + 1.
x^2 + 2x + 1 = (x + 1)(x + 1) = (x + 1)^2
So, the factored form of x^3 + 2x^2 + x is x(x + 1)^2.
By following these steps and applying the appropriate formulas, you can factor different types of expressions.