I have been doing sum of cubes and difference of cubes problems and I just cant figure out these questions...
Sum of Cubes Pattern
a3 + b3 = (a + b)(a2 - ab + b2)
Difference of Cubes Pattern
a3 - b3 = (a - b)(a2 + ab + b2
1. 64x^3 + 1
2. 1-125b^3
3. 8x^3y^6 + 27
4. a^3 b^6 – b^3
5. 27x^3 – 8y^3
1. To factorize 64x^3 + 1 using the sum of cubes pattern, we look for the cube roots of each term.
64x^3 can be expressed as (4x)^3 and 1 can be expressed as 1^3.
Applying the sum of cubes pattern, we get:
64x^3 + 1 = (4x + 1)((4x)^2 - (4x)(1) + 1)
Simplifying further:
64x^3 + 1 = (4x + 1)(16x^2 - 4x + 1)
2. To factorize 1 - 125b^3 using the difference of cubes pattern, we look for the cube roots of each term.
1 can be expressed as 1^3 and 125b^3 can be expressed as (5b)^3.
Applying the difference of cubes pattern, we get:
1 - 125b^3 = (1 - 5b)(1^2 + (1)(5b) + (5b)^2)
Simplifying further:
1 - 125b^3 = (1 - 5b)(1 + 5b + 25b^2)
3. To factorize 8x^3y^6 + 27 using the sum of cubes pattern, we look for the cube roots of each term.
8x^3y^6 can be expressed as (2xy^2)^3 and 27 can be expressed as 3^3.
Applying the sum of cubes pattern, we get:
8x^3y^6 + 27 = (2xy^2 + 3)((2xy^2)^2 - (2xy^2)(3) + 3^2)
Simplifying further:
8x^3y^6 + 27 = (2xy^2 + 3)(4x^2y^4 - 6xy^2 + 9)
4. To factorize a^3b^6 - b^3, we can first factor out the common factor b^3:
a^3b^6 - b^3 = b^3(a^3b^3 - 1)
Now we have a difference of cubes pattern, where a^3b^3 can be expressed as (ab)^3 and 1 can be expressed as 1^3.
Applying the difference of cubes pattern, we get:
a^3b^6 - b^3 = b^3(ab - 1)((ab)^2 + (ab)(1) + 1)
Simplifying further:
a^3b^6 - b^3 = b^3(ab - 1)(a^2b^2 + ab + 1)
5. To factorize 27x^3 - 8y^3 using the difference of cubes pattern, we look for the cube roots of each term.
27x^3 can be expressed as (3x)^3 and 8y^3 can be expressed as (2y)^3.
Applying the difference of cubes pattern, we get:
27x^3 - 8y^3 = (3x - 2y)((3x)^2 + (3x)(2y) + (2y)^2)
Simplifying further:
27x^3 - 8y^3 = (3x - 2y)(9x^2 + 6xy + 4y^2)
Sure! Let's go through each question step by step and use the patterns for sum of cubes and difference of cubes to solve them.
Question 1: 64x^3 + 1
To apply the sum of cubes pattern, we need to rewrite the expression as (a^3 + b^3). In this case, we have (64x^3 + 1) which can be understood as (4x)^3 + 1^3.
Now we can use the sum of cubes pattern: a^3 + b^3 = (a + b)(a^2 - ab + b^2). In this case, a = 4x, b = 1. So, substituting these values into the pattern, we have:
(4x)^3 + 1^3 = (4x + 1)(16x^2 - 4x + 1).
Therefore, the answer is (4x + 1)(16x^2 - 4x + 1).
Question 2: 1 - 125b^3
To apply the difference of cubes pattern, we need to rewrite the expression as (a^3 - b^3). In this case, we have 1 - 125b^3, which can be understood as 1^3 - (5b)^3.
Now we can use the difference of cubes pattern: a^3 - b^3 = (a - b)(a^2 + ab + b^2). In this case, a = 1, b = 5b. So, substituting these values into the pattern, we have:
1^3 - (5b)^3 = (1 - 5b)(1^2 + (1)(5b) + (5b)^2).
Simplifying further, we get:
(1 - 5b)(1 + 5b + 25b^2).
Therefore, the answer is (1 - 5b)(1 + 5b + 25b^2).
Question 3: 8x^3y^6 + 27
In this case, we don't have a straightforward sum or difference of cubes form. We need to factor out common terms first.
8x^3y^6 + 27 can be rewritten as (2xy^2)^3 + 3^3.
Now we can use the sum of cubes pattern: a^3 + b^3 = (a + b)(a^2 - ab + b^2). In this case, a = 2xy^2, b = 3. So, substituting these values into the pattern, we have:
(2xy^2)^3 + 3^3 = (2xy^2 + 3)(4x^2y^4 - 6xy^2 + 9).
Therefore, the answer is (2xy^2 + 3)(4x^2y^4 - 6xy^2 + 9).
Question 4: a^3b^6 - b^3
In this case, we have the difference of cubes, but only for the first term. So we can rewrite the expression as (a^3b^6 - b^3) = (a^3b^6 - b^3b^3).
Now we can use the difference of cubes pattern: a^3 - b^3 = (a - b)(a^2 + ab + b^2). In this case, a = ab^2, b = b. So, substituting these values into the pattern, we have:
(a^3b^6 - b^3b^3) = (ab^2 - b)(a^2b^4 + ab^3 + b^6).
Therefore, the answer is (ab^2 - b)(a^2b^4 + ab^3 + b^6).
Question 5: 27x^3 - 8y^3
Similar to question 4, in this case, we have the difference of cubes, but only for the first term. So, we can rewrite the expression as (27x^3 - 8y^3) = (3x)^3 - (2y)^3.
Now we can use the difference of cubes pattern: a^3 - b^3 = (a - b)(a^2 + ab + b^2). In this case, a = 3x, b = 2y. So, substituting these values into the pattern, we have:
(3x)^3 - (2y)^3 = (3x - 2y)((3x)^2 + (3x)(2y) + (2y)^2).
Simplifying further, we get:
(3x - 2y)(9x^2 + 6xy + 4y^2).
Therefore, the answer is (3x - 2y)(9x^2 + 6xy + 4y^2).
I hope this helps you with solving sum of cubes and difference of cubes problems! If you have any more questions, feel free to ask.