1.3x^2+17x+10
2.5y^8-125
3.a^2-2ab-15b^2
*Please help me.You have to factor by completing.
Let's look at #1:
3x^2 + 17x + 10
We have 2 and 5 as factors of the last term.
What combinations can we do to arrive at the middle term?
(3x + ?)(x + ?)
Let's try (3x + 5)(x + 2), which will be 3x^2 + 11x + 10. This is not what we want.
Let's try (3x + 2)(x + 5), which will be 3x^2 + 17x + 10. This IS what we want; therefore, we have found the factors of this trinomial.
Here's a few hints for the other two.
Hint for #2: First factor out 5. Then factor what's left.
Hint for #3: First factor (a - 5b). Can you determine the second factor?
I hope this will help.
To factor the second expression, 5y^8 - 125:
First, notice that both terms divisible by 5. We can factor out 5 from both terms:
5(y^8 - 25)
Now, let's focus on factoring the expression inside the parentheses, y^8 - 25. This is a difference of squares, since y^8 can be written as (y^4)^2 and 25 can be written as 5^2.
So, we can rewrite the expression as:
5((y^4)^2 - 5^2)
Using the formula for a difference of squares, which states that a^2 - b^2 = (a + b)(a - b), we can factor the expression further:
5(y^4 + 5)(y^4 - 5)
Therefore, the factored form of 5y^8 - 125 is 5(y^4 + 5)(y^4 - 5).
Now, let's move on to the third expression, a^2 - 2ab - 15b^2:
This expression does not have any common factors that can be factored out. So, we need to look for the factors that can be multiplied to give the middle term (-2ab) and the last term (-15b^2), while adding up to give the coefficient of the middle term (-2b).
In this case, the factors of -15b^2 that can be added to give -2b are -5b and 3b. So, we can rewrite the expression as:
(a^2 - 5ab + 3ab - 15b^2)
Now, we group the terms in pairs and factor out the common factors:
a(a - 5b) + 3b(a - 5b)
Notice that both terms have a common factor of (a - 5b). We can factor this out:
(a - 5b)(a + 3b)
Therefore, the factored form of a^2 - 2ab - 15b^2 is (a - 5b)(a + 3b).
I hope this helps you understand how to factor by completing.