222 - Steve, Wednesday, October 22, 2014 at 12:02am
I assume you want to factor the expressions?
a^4b + a^2b^3 = a^2b(a^2+b^2)
z^2 + 18z + 45 = (z+3)(z+15)
•Math 222 - Monica, Wednesday, October 22, 2014 at 12:10am
Thank you Steve and can you please explain how you got both answers, so that I can get a better understanding. Thanks
in the first, each term contains a power of a and a power of b. You need to find the highest power of a and b that divides each term. That would look like
a^2 a^2 b + a^2 b^2 b
Now factor out the a^2 b, leaving the indicated solution. Recall that
a^m a^n = a^(m+n)
Review the properties of exponents, and the distributive law.
For the second one, recall that
(z+a)(z+b) = z^2 + (a+b)z + ab
So, you want two numbers that are factors of 45, and add up to 18.
45 has factors of
1,3,5,9,15,45
Take the factors in pairs, working in from both ends, until you find the pair that adds up to 18.
Skill in factoring comes only from experience. Do lots of these. You can use sites like wolframalpha.com to check your results.
Sure, I'd be happy to explain how to factor those expressions.
For the expression a^4b + a^2b^3, notice that both terms have a common factor of ab. So we can factor out the ab, giving us:
a^4b + a^2b^3 = ab(a^4 + a^2b^2)
Now, we can see that the remaining terms inside the parentheses, a^4 + a^2b^2, are perfect squares. Specifically, a^4 is the square of a^2, and a^2b^2 is the square of ab. So we can write that expression as:
a^4 + a^2b^2 = (a^2)^2 + (ab)^2
By using the formula for the sum of squares (a^2 + b^2 = (a + b)(a - b)), we can factor the expression further:
(a^2)^2 + (ab)^2 = (a^2 + ab)(a^2 - ab)
Therefore, the fully factored expression is:
a^4b + a^2b^3 = ab(a^2 + ab)(a^2 - ab)
Now let's move on to the expression z^2 + 18z + 45. This expression doesn't have any common factors, so we need to look for two numbers that multiply to give 45 and add up to 18. Those numbers are 3 and 15.
Therefore, we can write:
z^2 + 18z + 45 = (z + 3)(z + 15)
So the fully factored expression is:
z^2 + 18z + 45 = (z + 3)(z + 15)
I hope this explanation helps you understand how to factor these expressions! Let me know if you have any further questions.