Completely factor the following expressions. Please show work/explain
t^2+4tv+4v2
4x^2-8x-12+6x
144-9p^2
5c^2-24cd-5d^2
w^2-17w+42
256z^2-4-192z^2+3
2a^2c^3-14bc^3+32c^3d^2
35g^2+6g-9
3j^3-51j^2+210j
looks like ur trying to pass off ur homework on us...
no, the homework was 50 questions, these are just the ones I've tried to do several times and failed miserably at.
1. a perfect square --- easy
2. use grouping
3. difference of squares
4. (5x + d)(x - 5d)
5. I see two numbers that when multiplied give me 42 , and when added will give me -17,
Can you see : -14, -3
6. simplify to get a difference of squares
7. ?? all I can see is a common factor of 2
8. take out common factor of 3j, then work on the quadratic
To completely factor the given expressions, we'll look for common factors and apply different factoring techniques such as factoring by grouping, difference of squares, or perfect square trinomials. Let's factor each expression one by one:
1. t^2 + 4tv + 4v^2:
To factor this expression, we notice that it is a perfect square trinomial. The formula for factoring a perfect square trinomial is (a + b)^2 = a^2 + 2ab + b^2. In this case, t^2 + 4tv + 4v^2 can be factored as (t + 2v)^2.
2. 4x^2 - 8x - 12 + 6x:
First, group the terms: (4x^2 - 8x) + (-12 + 6x).
Now factor out the common factor from each group: 4x(x - 2) + 6(x - 2).
Now, observe that we have a common factor (x - 2) in both terms, so we can factor it out: (x - 2)(4x + 6).
3. 144 - 9p^2:
The given expression does not contain any common factors, but it is a difference of squares. The formula for factoring a difference of squares is a^2 - b^2 = (a + b)(a - b). In this case, we can factor the expression as (12 + 3p)(12 - 3p).
4. 5c^2 - 24cd - 5d^2:
This expression does not have any common factors, so we'll use the factoring by grouping method.
First, group the terms: (5c^2 - 5d^2) + (-24cd).
Now factor out the common factor from each group: 5(c^2 - d^2) - 24cd.
Next, apply the difference of squares formula on (c^2 - d^2): (c + d)(c - d).
Now we have (5(c + d)(c - d)) - 24cd.
5. w^2 - 17w + 42:
To factor this expression, we'll look for two numbers whose product is 42 and sum is -17. The numbers that satisfy this are -2 and -15.
Therefore, we can factor the expression as (w - 2)(w - 15).
6. 256z^2 - 4 - 192z^2 + 3:
First, group the like terms: (256z^2 - 192z^2) + (-4 + 3).
Now combine like terms: 64z^2 - 1.
Since this expression does not have any common factors and cannot be factored further, it remains as 64z^2 - 1.
7. 2a^2c^3 - 14bc^3 + 32c^3d^2:
First, group the like terms: (2a^2c^3 - 14bc^3) + (32c^3d^2).
Now factor out the common factor from each group: 2c^3(a^2 - 7b) + 32c^3d^2.
We can further factor out the common factor 2c^3: 2c^3(a^2 - 7b + 16d^2).
8. 35g^2 + 6g - 9:
To factor this expression, we'll look for two numbers whose product is -9 and sum is 6. The numbers that satisfy this are 3 and -3.
Therefore, we can factor the expression as (5g - 3)(7g + 3).
9. 3j^3 - 51j^2 + 210j:
First, factor out the common factor j: j(3j^2 - 51j + 210).
To further factor the quadratic expression, we'll look for two numbers whose product is 210 and sum is -51. The numbers that satisfy this are -6 and -35.
Therefore, we can factor the expression as j(3j - 6)(j - 35).
Remember, factoring expressions involves identifying common factors, using factoring techniques like the difference of squares or factoring by grouping, and finding suitable factors for quadratic expressions.