Answer
To factor and simplify the given expressions:
1. **Investigation**
- \( x^{2} - y^{2} = (x - y)(x + y) \)
- \( 625a^{4} - 81b^{8} = (5a - 3b^{2})(5a + 3b^{2})(25a^{2} + 9b^{4}) \)
- \( 4(x - y)^{2} - 9(x + y)^{2} = -(5x + y)(x + 5y) \)
2. **Question 2**
- \( x^{2} + 2xy + y^{2} = (x + y)^{2} \)
- \( x^{2} - 8x + 16 = (x - 4)^{2} \)
- \( 16a^{4} - 24a^{2}b^{3} + 9b^{6} = (3b^{3} - 4a^{2})^{2} \)
3. **Question 3**
- \( x^{3} + y^{3} = (x + y)(x^{2} - xy + y^{2}) \)
- \( 8a^{6} + 729 = (2a^{2} + 9)(4a^{4} - 18a^{2} + 81) \)
- \( (3x + y)^{3} + (3x - y)^{3} = 18x(3x^{2} + y^{2}) \)
**General Statement:**
Factorization involves recognizing patterns such as perfect squares, sums and differences of cubes, and common factors to simplify expressions.
Solution
Factor the expression by following steps:
- step0: Factor:
\(x^{2}+2xy+y^{2}\)
- step1: Factor the expression:
\(\left(x+y\right)^{2}\)
Factor the expression \( 625 a^{4}-81 b^{8} \).
Factor the expression by following steps:
- step0: Factor:
\(625a^{4}-81b^{8}\)
- step1: Rewrite the expression:
\(\left(25a^{2}\right)^{2}-\left(9b^{4}\right)^{2}\)
- step2: Factor the expression:
\(\left(25a^{2}-9b^{4}\right)\left(25a^{2}+9b^{4}\right)\)
- step3: Evaluate:
\(\left(5a-3b^{2}\right)\left(5a+3b^{2}\right)\left(25a^{2}+9b^{4}\right)\)
Factor the expression \( (3 x+y)^{3}+(3 x-y)^{3} \).
Factor the expression by following steps:
- step0: Factor:
\(\left(3x+y\right)^{3}+\left(3x-y\right)^{3}\)
- step1: Factor the expression:
\(\left(3x+y+3x-y\right)\left(\left(3x+y\right)^{2}-\left(3x+y\right)\left(3x-y\right)+\left(3x-y\right)^{2}\right)\)
- step2: Evaluate:
\(\left(3x+y+3x-y\right)\left(\left(3x+y\right)^{2}+\left(-3x-y\right)\left(3x-y\right)+\left(3x-y\right)^{2}\right)\)
- step3: Calculate:
\(6x\left(\left(3x+y\right)^{2}+\left(-3x-y\right)\left(3x-y\right)+\left(3x-y\right)^{2}\right)\)
- step4: Calculate:
\(6x\left(9x^{2}+3y^{2}\right)\)
- step5: Factor the expression:
\(6x\times 3\left(3x^{2}+y^{2}\right)\)
- step6: Simplify the expression:
\(18x\left(3x^{2}+y^{2}\right)\)
Factor the expression \( 16 a^{4}-24 a^{2} b^{3}+9 b^{6} \).
Factor the expression by following steps:
- step0: Factor:
\(16a^{4}-24a^{2}b^{3}+9b^{6}\)
- step1: Factor the expression:
\(\left(3b^{3}-4a^{2}\right)^{2}\)
Factor the expression \( 4(x-y)^{2}-9(x+y)^{2} \).
Factor the expression by following steps:
- step0: Factor:
\(4\left(x-y\right)^{2}-9\left(x+y\right)^{2}\)
- step1: Factor the expression:
\(\left(2\left(x-y\right)+3\left(x+y\right)\right)\left(2\left(x-y\right)-3\left(x+y\right)\right)\)
- step2: Calculate:
\(\left(5x+y\right)\left(-x-5y\right)\)
- step3: Factor the expression:
\(\left(5x+y\right)\left(-\left(x+5y\right)\right)\)
- step4: Calculate:
\(\left(5x+y\right)\left(-1\right)\left(x+5y\right)\)
- step5: Simplify the expression:
\(-\left(5x+y\right)\left(x+5y\right)\)
Factor the expression \( x^{2}-8 x+16 \).
Factor the expression by following steps:
- step0: Factor:
\(x^{2}-8x+16\)
- step1: Factor the expression:
\(\left(x-4\right)^{2}\)
Factor the expression \( x^{2}-y^{2} \).
Factor the expression by following steps:
- step0: Factor:
\(x^{2}-y^{2}\)
- step1: Factor the expression:
\(\left(x-y\right)\left(x+y\right)\)
Factor the expression \( x^{3}+y^{3 \).
Factor the expression by following steps:
- step0: Factor:
\(x^{3}+y^{3}\)
- step1: Factor the expression:
\(\left(x+y\right)\left(x^{2}-xy+y^{2}\right)\)
Factor the expression \( 8 a^{6}+729 \).
Factor the expression by following steps:
- step0: Factor:
\(8a^{6}+729\)
- step1: Rewrite the expression:
\(\left(2a^{2}\right)^{3}+9^{3}\)
- step2: Factor the expression:
\(\left(2a^{2}+9\right)\left(\left(2a^{2}\right)^{2}-2a^{2}\times 9+9^{2}\right)\)
- step3: Evaluate:
\(\left(2a^{2}+9\right)\left(4a^{4}-18a^{2}+81\right)\)
Let's factor and simplify the given expressions step by step.
### Investigation
1.1.1 \( x^{2}-y^{2} \)
This is a difference of squares, which can be factored as:
\[
x^{2}-y^{2} = (x-y)(x+y)
\]
1.1.2 \( 625 a^{4}-81 b^{8} \)
This is also a difference of squares:
\[
625 a^{4}-81 b^{8} = (25a^{2}-9b^{4})(25a^{2}+9b^{4})
\]
Now, we can factor \( 25a^{2}-9b^{4} \) further:
\[
25a^{2}-9b^{4} = (5a-3b^{2})(5a+3b^{2})
\]
Thus, the complete factorization is:
\[
625 a^{4}-81 b^{8} = (5a-3b^{2})(5a+3b^{2})(25a^{2}+9b^{4})
\]
1.1.3 \( 4(x-y)^{2}-9(x+y)^{2} \)
This can be rewritten as:
\[
4(x-y)^{2}-9(x+y)^{2} = -(9(x+y)^{2}-4(x-y)^{2}) = -(5x+y)(x+5y)
\]
### General Statement for Investigation
To factor expressions like these, look for patterns such as the difference of squares, perfect squares, or common factors. Recognizing these patterns allows for systematic factorization.
---
### Question 2
2.1.1 \( x^{2}+2xy+y^{2} \)
This is a perfect square trinomial:
\[
x^{2}+2xy+y^{2} = (x+y)^{2}
\]
2.1.2 \( x^{2}-8x+16 \)
This is also a perfect square trinomial:
\[
x^{2}-8x+16 = (x-4)^{2}
\]
2.1.3 \( 16 a^{4}-24 a^{2} b^{3}+9 b^{6} \)
This can be factored as:
\[
16 a^{4}-24 a^{2} b^{3}+9 b^{6} = (3b^{3}-4a^{2})^{2}
\]
### General Statement for Question 2
Identifying perfect square trinomials is key in factorization. Look for patterns where the expression can be rewritten as a square of a binomial.
---
### Question 3
3.1.1 \( x^{3}+y^{3} \)
This is a sum of cubes:
\[
x^{3}+y^{3} = (x+y)(x^{2}-xy+y^{2})
\]
3.1.2 \( 8 a^{6}+729 \)
This is also a sum of cubes:
\[
8 a^{6}+729 = (2a^{2}+9)(4a^{4}-18a^{2}+81)
\]
3.1.3 \( (3x+y)^{3}+(3x-y)^{3} \)
This can be factored using the sum of cubes formula:
\[
(3x+y)^{3}+(3x-y)^{3} = 18x(3x^{2}+y^{2})
\]
### General Statement for Question 3
Recognizing sums and differences of cubes is essential for factorization. Use the formulas for sum and difference of cubes to simplify these expressions.
---
In summary, the key to factorization lies in recognizing patterns such as perfect squares, sums and differences of cubes, and common factors.
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