Certainly! Let's go through each part of the problem step by step.
### Part 1: Factoring the given expressions
#### Expression 1
Given: [tex]\( 3a^2 - 6a^2b \)[/tex]
We can factor out the common term [tex]\( 3a^2 \)[/tex]:
[tex]\[
3a^2 - 6a^2b = 3a^2(1 - 2b)
\][/tex]
Thus, the factored form is:
[tex]\[
3a^2(1 - 2b)
\][/tex]
#### Expression 2
Given: [tex]\( 2x^3 + 4x^2 + 10x - 5y + 35 \)[/tex]
Notice that [tex]\( 2x^3, 4x^2, 10x \)[/tex] involve [tex]\( x \)[/tex], while [tex]\(-5y + 35\)[/tex] do not. We can first group the terms with [tex]\( x \)[/tex] and the constant terms separately:
[tex]\[
(2x^3 + 4x^2 + 10x) + (-5y + 35)
\][/tex]
We can further factor the grouped terms:
For [tex]\( 2x^3 + 4x^2 + 10x \)[/tex], factor out the common factor [tex]\( 2x \)[/tex]:
[tex]\[
2x(x^2 + 2x + 5)
\][/tex]
For [tex]\(-5y + 35\)[/tex], factor out the common factor [tex]\(-5\)[/tex]:
[tex]\[
-5(y - 7)
\][/tex]
Thus, the expression can be written as:
[tex]\[
2x(x^2 + 2x + 5) - 5(y - 7)
\][/tex]
#### Expression 3
Given: [tex]\( 3x^2y - 5 \)[/tex]
This expression cannot be factored further as it is already in its simplest form:
[tex]\[
3x^2y - 5
\][/tex]
### Part 2: Writing as a polynomial
#### Polynomial 1
Given: [tex]\((x + 3)(x + 7)\)[/tex]
We can expand this polynomial:
[tex]\[
(x + 3)(x + 7) = x^2 + 7x + 3x + 21 = x^2 + 10x + 21
\][/tex]
So, the expanded polynomial is:
[tex]\[
x^2 + 10x + 21
\][/tex]
#### Polynomial 2
Given: [tex]\((x - 4)^2\)[/tex]
We can expand this polynomial using the formula [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]:
[tex]\[
(x - 4)^2 = x^2 - 2 \cdot x \cdot 4 + 4^2 = x^2 - 8x + 16
\][/tex]
So, the expanded polynomial is:
[tex]\[
x^2 - 8x + 16
\][/tex]
In summary, the polynomials after expansion are:
[tex]\[
(x + 3)(x + 7) = x^2 + 10x + 21
\][/tex]
[tex]\((x - 4)^2 = x^2 - 8x + 16\)[/tex]
And factoring discussed:
1. [tex]\( 3a^2 - 6a^2b \)[/tex] simplifies to [tex]\( 3a^2(1 - 2b) \)[/tex]
2. [tex]\( 2x^3 + 4x^2 + 10x - 5y + 35 \)[/tex] simplifies to [tex]\( 2x(x^2 + 2x + 5) - 5(y - 7) \)[/tex]
3. [tex]\( 3x^2y - 5 \)[/tex] remains [tex]\( 3x^2y - 5 \)[/tex]
