Factorize:
1.) 64x^3-y^3
2.) x^4+10x^3+35x^2+50x+24
3.) x^3/8-64-3x^2+24x
1) an obvious difference of cubes
64x^3 - y^3 = (4x-y)(16x^2 + 4xy + y^2)
2) x^4+10x^3+35x^2+50x+24
I tried ±1, ±2, ±3
Found x=-1, x=-2, and x=-3 to work
so the first three factors would be
(x+1)(x+2)(x+3)(......)
By the "common sense theorem) the last factor would have to be (x+4)
3. trying different things ....
3. x^3/8-64-3x^2+24x
= (1/8)(x^3 - 512 - 24x^2 + 192x)
= (1/8)(x^3 - 24x^2 + 192x - 512)
noticed that 8^3 = 512, became suspicious it might be perfect cube, so
tried (x-8)^3 and found it to be x^3 - 24x^2 + 192x -512
so
x^3/8-64-3x^2+24x
= (1/8)(x-8)^3
1.) To factorize 64x^3 - y^3, we can use the identity for the difference of cubes. The difference of cubes formula states that a^3 - b^3 can be factored as (a - b)(a^2 + ab + b^2).
Using this formula, we can factorize 64x^3 - y^3 as follows:
64x^3 - y^3 = (4x - y)((4x)^2 + (4x)(y) + (y)^2)
= (4x - y)(16x^2 + 4xy + y^2)
So, the factored form of 64x^3 - y^3 is (4x - y)(16x^2 + 4xy + y^2).
2.) To factorize x^4 + 10x^3 + 35x^2 + 50x + 24, we can use the factoring by grouping method. This involves grouping the terms into pairs and factoring out the greatest common factor from each pair.
Grouping the terms, we have:
(x^4 + 10x^3) + (35x^2 + 50x) + 24
Now, we can factor out the greatest common factor from each pair:
x^3(x + 10) + 5x(7x + 10) + 24
Next, we can factor out the common factors from the grouped terms:
x^3(x + 10) + 5x(7x + 10) + 24
= x^3(x + 10) + 5x(7x + 10) + 24(1)
Notice that we now have a common factor of (x + 10) between the first two terms and a common factor of (7x + 10) between the second and third terms.
Factoring out the common factors:
(x + 10)(x^3 + 5x) + (7x + 10)(5x + 24)
So, the factored form of x^4 + 10x^3 + 35x^2 + 50x + 24 is (x + 10)(x^3 + 5x) + (7x + 10)(5x + 24).
3.) To factorize x^3/8 - 64 - 3x^2 + 24x, we can factor out the greatest common factor.
First, let's rewrite the expression with common denominators:
(x^3/8) - (512/8) - 3x^2 + 24x
Simplifying the expression, we have:
(x^3 - 512)/8 - 3x^2 + 24x
We can see that the greatest common factor is (x - 8), so we can factor it out from the expression:
(x - 8)((x^3 - 512)/8 - 24x)
Now, let's focus on the remaining terms inside the parentheses. Notice that (x^3 - 512) is a difference of cubes, which can be factored using the same formula as in the first question.
(x - 8)((x - 8)(x^2 + 8x + 64)/8 - 24x)
Further simplifying, we have:
(x - 8)^2(x^2 + 8x + 64)/8 - 24x(x - 8)
So, the factored form of x^3/8 - 64 - 3x^2 + 24x is (x - 8)^2(x^2 + 8x + 64)/8 - 24x(x - 8).
To factorize the given expressions, we need to find the common factors and then apply methods like factoring by grouping or using known formulas for factorization.
1.) 64x^3 - y^3:
To factorize this expression, we can use the difference of cubes formula, which states that a^3 - b^3 = (a - b)(a^2 + ab + b^2). In this case, a is 4x and b is y.
Therefore, the factorization of 64x^3 - y^3 is:
64x^3 - y^3 = (4x - y)(16x^2 + 4xy + y^2)
2.) x^4 + 10x^3 + 35x^2 + 50x + 24:
To factorize this expression, we need to find the common factors if there are any. In this case, we don't have any common factors among all the terms.
We can apply factoring by grouping to factorize it. This technique involves grouping terms in pairs and factoring out the common factors.
x^4 + 10x^3 + 35x^2 + 50x + 24 can be grouped as:
(x^4 + 10x^3) + (35x^2 + 50x) + 24
Now, we factor out the common factors from each group:
x^3(x + 10) + 5x(7x + 10) + 24
Now, we have a common factor of (x + 10). Factoring that out, we get:
(x + 10)(x^3 + 5x^2 + 7x + 24)
3.) x^3/8 - 64 - 3x^2 + 24x:
To factorize this expression, we can proceed in steps:
First, we can factor out the common factors, which is 8 in this case:
8(x^3/8 - 8 - (3/8)x^2 + 3x)
Next, we can factor by grouping:
8(x^3/8 - (3/8)x^2) - 8(8 - 3x)
Factor out the common factors from each group:
8(x^2(x/8 - 3/8)) - 8(-1)(3x - 8)
Now, we can simplify further:
8(x/8 - 3/8)(x^2 + 3)
The final factorization of x^3/8 - 64 - 3x^2 + 24x is:
8(x/8 - 3/8)(x^2 + 3)