Rewrite each polynomial as the product of two binomials.
1. y^2 + 6y + 9
2. 25x^2 - 10x + 1
3. 36 - y^2
I don't understand how to do these problems, especially what the direction is telling me to do. Someone help me please?
They want you to factor the polynomials. For example,
36 - y^2 = (6 + y)(6 - y)
The other two are perfect squares of monomials. Make an effort to solve them. Someone will critique your efforts
(a+b)^2 = a^2 + 2 a b + b^2
(a-b)^2 = a^2 - 2 a b + b^2
Damon has given you some useful hints to solve 1. and 2. For example, in #2, let 5x = a and 1 = b and see what you get.
No problem! I'm here to help you understand how to rewrite these polynomials as the product of two binomials.
To rewrite a polynomial as the product of two binomials, you will need to use a method called factoring. Factoring involves breaking down the polynomial into two binomial expressions by finding two binomials that, when multiplied together, will result in the original polynomial.
Now, let's work on the first polynomial: y^2 + 6y + 9.
Step 1: Look for common factors.
In this case, there are no common factors among the terms.
Step 2: Identify the pattern of the polynomial.
The polynomial y^2 + 6y + 9 appears to fit the pattern of a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored as the square of a binomial.
Step 3: Apply the perfect square trinomial formula.
The perfect square trinomial formula is: (a + b)^2 = a^2 + 2ab + b^2
To factor y^2 + 6y + 9, we can use the formula to determine that:
(a + b)^2 = (y + 3)^2 = y^2 + 6y + 9
Therefore, y^2 + 6y + 9 can be rewritten as (y + 3)(y + 3) or (y + 3)^2.
Let's move on to the second polynomial: 25x^2 - 10x + 1.
Step 1: Look for common factors.
Again, there are no common factors among the terms.
Step 2: Identify the pattern of the polynomial.
The polynomial 25x^2 - 10x + 1 does not fit the pattern of a perfect square trinomial or any other known factoring pattern. Therefore, we need to use a different approach.
Step 3: Factor by using trial and error or the quadratic formula.
In this case, we can factor 25x^2 - 10x + 1 by applying the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
For the polynomial 25x^2 - 10x + 1, the quadratic formula yields:
x = (-(-10) ± √((-10)^2 - 4(25)(1)))/(2(25))
x = (10 ± √(100 - 100))/(50)
x = (10 ± √0)/(50)
x = (10 ± 0)/(50)
x = 10/50
x = 1/5
Since x = 1/5, we know that (x - 1/5) is a factor of the polynomial.
Therefore, 25x^2 - 10x + 1 can be rewritten as (x - 1/5)(25x - 5).
Finally, let's consider the third polynomial: 36 - y^2.
Step 1: Look for common factors.
There are no common factors among the terms.
Step 2: Identify the pattern of the polynomial.
The polynomial 36 - y^2 is in the form of a difference of squares. A difference of squares is when we have a^2 - b^2, which factors as (a + b)(a - b).
Step 3: Apply the difference of squares formula.
Using the difference of squares formula, we can write 36 - y^2 as (6 + y)(6 - y).
So, 36 - y^2 can be rewritten as (6 + y)(6 - y).
I hope this explanation helps you understand how to rewrite polynomials as the product of two binomials. If you have any further questions, feel free to ask.