Answer: 2p(2pq + 1)^2.
Please read the alternative explanation if you do not understand the first one. Also, there are other methods that work. Ultimetely, it is upto you to decide what works best.
The first step in factoring a Trinomial such as this is to see if all the terms are divisible by a common factor.
As you can see, they are all divisible by 2p.
Lets do our factoring:
8pq^2 + 8pq + 2p
= 2p (4q^2 + 4q + 1)
Now, let's look inside the parentheses, which has 3 terms inside of it. The three terms inside the parentheses are:
- 4q^2
- 4q
- 1
We need to multiply the first term's coefficient and the last term together.
First term's coefficient: 4
Last term: 1
4 * 1 = 4
Now, lets look at our middle term, 4q, with a coefficient of 4. We need to find 2 whole numbers that multiply to 4 and also add to 4.
Those numbers are 2 and 2.
So, we end up with:
2p (4q^2 + 4q + 1)
= 2p (4q^2 + 2q+2q + 1)
Next, we factor even further:
2p (4q^2 + 2q+2q + 1)
= 2p (2q(2q+1)+1(2q+1))
= 2p(2q+1)(2q+1)
= 2p(2q+1)^2
ALTERNATIVE EXPLANATION
Let's go through the factoring step by step:
The given polynomial is: 8pq^2 + 8pq + 2p
Step 1: Factor out the common term '2p' from all the terms:
We can take '2p' as a common factor from all the terms:
2p(4pq^2 + 4pq + 1)
Step 2: Focus on factoring the quadratic expression inside the parentheses:
The quadratic expression inside the parentheses is: 4pq^2 + 4pq + 1
To factor this quadratic expression, we look for two binomials in the form of (ax + b)(cx + d) that, when multiplied, give us the original quadratic expression. The first terms of the binomials should multiply to give us 4pq^2, which means 'a' and 'c' are both 2p (2p * 2p = 4p^2). The last terms of the binomials should multiply to give us 1, which means 'b' and 'd' are both 1 (1 * 1 = 1).
So, the binomials will be in the form of (2pq + 1)(2pq + 1).
Step 3: Write the factored form:
Now, we have factored the quadratic expression as (2pq + 1)(2pq + 1).
Step 4: Combine the common factor:
Finally, we can combine the common factor '2p' from Step 1 with the factored quadratic expression from Step 2:
2p(2pq + 1)(2pq + 1)
And that's the completely factored form of the original polynomial: 2p(2pq + 1)^2.
I hope this helped!
~~~Harsha~~~