Let's go through the process of factoring each of the given polynomials step by step:
### 1. [tex]\( y^2 - 10y \)[/tex]
We start by factoring out the greatest common factor (GCF) from the polynomial:
[tex]$ y^2 - 10y = y(y - 10) $[/tex]
Thus, the factored form is:
[tex]$ y(y - 10) $[/tex]
### 2. [tex]\( 4x^2 - 12x \)[/tex]
Again, we factor out the GCF from the polynomial:
[tex]$ 4x^2 - 12x = 4x(x - 3) $[/tex]
So the factored form is:
[tex]$ 4x(x - 3) $[/tex]
### 3. [tex]\( 6s^2 + 18s \)[/tex]
First, we factor out the GCF:
[tex]$ 6s^2 + 18s = 6s(s + 3) $[/tex]
Thus, the factored form is:
[tex]$ 6s(s + 3) $[/tex]
### 4. [tex]\( y^2 + 2y + 1 \)[/tex]
This quadratic polynomial resembles a perfect square trinomial. It can be written as:
[tex]$ y^2 + 2y + 1 = (y + 1)(y + 1) = (y + 1)^2 $[/tex]
Thus, the factored form is:
[tex]$ (y + 1)^2 $[/tex]
### 5. [tex]\( x^2 - 3x - 10 \)[/tex]
For this quadratic polynomial, we look for two numbers that multiply to [tex]\(-10\)[/tex] (the constant term) and add up to [tex]\(-3\)[/tex] (the coefficient of the linear term). These numbers are [tex]\(-5\)[/tex] and [tex]\(2\)[/tex]. Therefore, we can factor the polynomial as follows:
[tex]$ x^2 - 3x - 10 = (x - 5)(x + 2) $[/tex]
So the factored form is:
[tex]$ (x - 5)(x + 2) $[/tex]
### Summary
Here are the factored forms of the given polynomials:
1. [tex]\( y(y - 10) \)[/tex]
2. [tex]\( 4x(x - 3) \)[/tex]
3. [tex]\( 6s(s + 3) \)[/tex]
4. [tex]\( (y + 1)^2 \)[/tex]
5. [tex]\( (x - 5)(x + 2) \)[/tex]
