Question

factor the following polynomials based on the examples shown in class. you will end up with either a binomial squared or two binomials depending on the special case presented there may or may not be a common factor. show your work!!!

1. ( a^2 - 4 )
2. ( n^2 - 64 )
3. ( 81 - x^2 )
4. ( 9b^2 - 100 )
5. ( x^2 - 81y^2 )
6. ( 81n^4 - 25 )
7. ( 49m^4 - 16 )
8. ( k^6 - 16 )
9. ( 64m^4 - 9n^2 )

Explanation:

Step1: Recognize difference of squares

All polynomials follow $a^2 - b^2 = (a-b)(a+b)$.

Step2: Factor each polynomial

1. $a^2 - 4 = a^2 - 2^2 = (a-2)(a+2)$
2. $n^2 - 64 = n^2 - 8^2 = (n-8)(n+8)$
3. $81 - x^2 = 9^2 - x^2 = (9-x)(9+x)$
4. $9b^2 - 100 = (3b)^2 - 10^2 = (3b-10)(3b+10)$
5. $x^2 - 81y^2 = x^2 - (9y)^2 = (x-9y)(x+9y)$
6. $81n^4 - 25 = (9n^2)^2 - 5^2 = (9n^2-5)(9n^2+5)$
7. $49m^4 - 16 = (7m^2)^2 - 4^2 = (7m^2-4)(7m^2+4)$
8. $k^6 - 16 = (k^3)^2 - 4^2 = (k^3-4)(k^3+4)$
9. $64m^4 - 9n^2 = (8m^2)^2 - (3n)^2 = (8m^2-3n)(8m^2+3n)$

Answer:

1. $(a-2)(a+2)$
2. $(n-8)(n+8)$
3. $(9-x)(9+x)$
4. $(3b-10)(3b+10)$
5. $(x-9y)(x+9y)$
6. $(9n^2-5)(9n^2+5)$
7. $(7m^2-4)(7m^2+4)$
8. $(k^3-4)(k^3+4)$
9. $(8m^2-3n)(8m^2+3n)$