Can you check the following problems?
Factor the following.
1. x^2 + 9x + 18
= (x + 6)(x + 3)
8. 3x^3 - 24x^2 - 60x
= 3x(x^2 - 8x - 20)
= 3x(x-10)(x + 2)
9. 5x^4 - 5x^3 - 30x^2
= 5x^2(x^2 - x - 6)
= 5x^2(x - 3)(x + 2)
10. 6x^2 - x - 2
= (x - 4/6)(x + 3/6)
= (x -2/3) (x +1/2)
= (3x - 2)(2x + 1)
17. (m^4 - 81)
= (m^2 + 9)(m^2 - 9)
= (m^2 + 9)(m + 3)(m - 3)
18. 4b^2 - 400
= 4(b^2 - 100)
= 4(b + 10)(b - 10)
22. x^2 - 2x - 3 = 0
(x - 3)(x + 1) = 0
x - 3 = 0 x + 1 = 0
x = 3, x = -1
23. 6b^3 - 28b^2 + 30b = 0
2b(3b^2 - 14b + 15) = 0
2b(b - 9)(b - 5) = 0
2b (b-9/3)(b-5/3) = 0
2b(b-3)(3b-5) = 0
2b = 0, b-3 = 0, 3b - 5 = 0
b = 0, b = 3, b = 3/5
24. 5m^2 + 11m - 12 = 0
(m + 15)(m - 4) = 0
(m +15/5)(m -4/5) = 0
(m + 3)(5m - 4) = 0
m + 3 = 0, 5m - 4 = 0
m - -3, m = 4/5
25. 27x^2 - 12 = 0
3(9x^2 - 4) = 0
3(3x - 2)(3x + 2) = 0
3 != 0, 3x - 2 = 0, 3x + 2 = 0
3 != 0, x = 2/3, x = -2/3
all correct. Good job.
thanks for checking!!
To factor the given expressions, follow these steps:
1. Start by checking if the expression has any common factors. If it does, factor them out.
2. Then, look for patterns or specific techniques to factor the remaining expression.
3. Finally, check if any of the factors can be further factored.
Let's go through each problem:
1. x^2 + 9x + 18
To factor this quadratic expression, we need to find two numbers that multiply to 18 and add up to 9. The numbers are 6 and 3. Therefore, the expression can be factored as (x + 6)(x + 3).
8. 3x^3 - 24x^2 - 60x
First, factor out the common factor, which is 3x:
3x(x^2 - 8x - 20)
Next, factor the quadratic expression in the parentheses. Find two numbers that multiply to -20 and add up to -8. The numbers are -10 and 2. Therefore, the expression can be factored as 3x(x-10)(x + 2).
9. 5x^4 - 5x^3 - 30x^2
First, factor out the common factor, which is 5x^2:
5x^2(x^2 - x - 6)
Next, factor the quadratic expression in the parentheses. Find two numbers that multiply to -6 and add up to -1. The numbers are -3 and 2. Therefore, the expression can be factored as 5x^2(x - 3)(x + 2).
10. 6x^2 - x - 2
To factor this quadratic expression, we need to find two numbers that multiply to -12 (6 * -2) and add up to -1. The numbers are -4 and 3. Therefore, the expression can be factored as (2x - 1)(3x + 2).
17. m^4 - 81
This expression is a difference of squares. It can be factored as (m^2 + 9)(m^2 - 9). Further factoring the difference of squares gives (m^2 + 9)(m + 3)(m - 3).
18. 4b^2 - 400
First, factor out the common factor, which is 4:
4(b^2 - 100)
Next, factor the quadratic expression in the parentheses. This is a difference of squares that can be factored as (b + 10)(b - 10).
Therefore, the expression can be factored as 4(b + 10)(b - 10).
22. x^2 - 2x - 3 = 0
To factor this quadratic equation, we need to find two numbers that multiply to -3 and add up to -2. The numbers are 3 and -1.
Therefore, the factored form of the equation is (x - 3)(x + 1) = 0.
Setting each factor equal to zero gives x = 3 and x = -1.
23. 6b^3 - 28b^2 + 30b = 0
First, factor out the common factor, which is 2b:
2b(3b^2 - 14b + 15) = 0
Next, factor the quadratic expression in the parentheses by factoring out a common factor or using the quadratic formula. The quadratic expression can be factored as (b - 9)(b - 5).
Therefore, the expression can be factored as 2b(b - 9)(b - 5).
Setting each factor equal to zero gives b = 0, b = 9, and b = 5/3.
The same steps can be followed for the remaining problems: 24 and 25.