Factor these polynomials completely. If the expression is not factorable, write prime.
1. 81 - 25a²
2. 44b⁴ - 44
3. 2(2k - 5)² - 18m⁴n⁴
4. 75p² - 3(2s + 7t)²
Share
Factor these polynomials completely. If the expression is not factorable, write prime.
1. 81 - 25a²
2. 44b⁴ - 44
3. 2(2k - 5)² - 18m⁴n⁴
4. 75p² - 3(2s + 7t)²
Sign Up to our social questions and Answers Engine to ask questions, answer people’s questions, and connect with other people.
Login to our social questions & Answers Engine to ask questions answer people’s questions & connect with other people.
Verified answer
Answer:
1. 81 - 25a²
This is a difference of squares, which can be factored as (9 - 5a)(9 + 5a).
Explanation: Difference of squares means that we have two perfect squares with a subtraction operation in between. In this case, we have 81 (which is 9²) and 25a² (which is (5a)²).
To factor a difference of squares, we simply need to find the square roots of both terms and write them as a subtraction and addition expression. So, the square root of 81 is 9, and the square root of 25a² is 5a.
Therefore, the factored form of 81 - 25a² is (9 - 5a)(9 + 5a).
2. 44b⁴ - 44
We can factor out 44 from both terms, resulting in 44(b⁴ - 1).
Explanation: To factor out a common factor from multiple terms, we look for the highest exponent or factor that occurs in all terms. In this case, both terms have a factor of 44.
For the second term, we need to recognize that 44 can also be expressed as 44 * 1. Then, we can apply the difference of squares factorization to b⁴ - 1.
The difference of squares expression b⁴ - 1 can be factored as (b² - 1)(b² + 1).
Therefore, the factored form of 44b⁴ - 44 is 44(b² - 1)(b² + 1).
3. 2(2k - 5)² - 18m⁴n⁴
We can factor out a common factor of 2, resulting in 2[(2k - 5)² - 9m⁴n⁴].
Explanation: To factor out a common factor from multiple terms, we look for the highest exponent or factor that occurs in all terms. In this case, both terms have a factor of 2.
Once we factor out 2, we are left with the expression (2k - 5)² - 9m⁴n⁴ to factorize further.
The expression (2k - 5)² is already a perfect square, so it cannot be factored any further.
The expression 9m⁴n⁴ is also a perfect square, as 9m⁴n⁴ = (3m²n²)².
Therefore, the factored form of 2(2k - 5)² - 18m⁴n⁴ is 2[(2k - 5)² - (3m²n²)²].
4. 75p² - 3(2s + 7t)²
The expression 75p² can be factored as 3p²(25 - 2s - 7t)(25 + 2s + 7t).
Explanation: To factor 75p², we look for common factors in its prime factorization. Here, both 75 and p² are factors. So, we can factor 75p² as 3p² * 25.
The expression (2s + 7t)² is already in a factored form because it is a perfect square.
Therefore, the factored form of 75p² - 3(2s + 7t)² is 3p²(25 - 2s - 7t)(25 + 2s + 7t).