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Factor the following expressions:

1. [tex]\( 10x + 10y + 10z \)[/tex]
2. [tex]\( 25x^2 y^3 = 25xy^3 \)[/tex]
3. [tex]\( 3x^2 y^{-4} z^5 + 15x^2 z^3 \)[/tex]


Sagot :

Certainly! Let's walk through the factoring of each expression step-by-step.

### 1. Factor [tex]\( 10(x + y + z_3) \)[/tex]

The given expression is [tex]\( 10x + 10y + 10z_3 \)[/tex]. Notice that each term has a common factor of [tex]\( 10 \)[/tex].

- Step 1: Identify the common factor
[tex]\[ 10x + 10y + 10z_3 \implies 10(x + y + z_3) \][/tex]

So, the factored form is:
[tex]\[ 10(x + y + z_3) \][/tex]

### 2. Factor [tex]\( 25x^2 y^3 - 25xy^3 \)[/tex]

The given expression is [tex]\( 25x^2 y^3 - 25xy^3 \)[/tex]. Notice that each term has a common factor of [tex]\( 25xy^3 \)[/tex].

- Step 1: Identify the common factor
[tex]\[ 25x^2 y^3 - 25xy^3 \implies 25xy^3(x - 1) \][/tex]

So, the factored form is:
[tex]\[ 25xy^3(x - 1) \][/tex]

### 3. Factor [tex]\( 3x^2 y^{-4} z^5 + 15x^2 z^3 \)[/tex]

The given expression is [tex]\( 3x^2 y^{-4} z^5 + 15x^2 z^3 \)[/tex]. Notice that each term has a common factor of [tex]\( 3x^2 z^3 \)[/tex].

- Step 1: Identify the common factor
[tex]\[ 3x^2 y^{-4} z^5 + 15x^2 z^3 \implies 3x^2 z^3(y^{-4} z^2 + 5) \][/tex]
[tex]\[ \implies 3x^2 z^3(\frac{z^2}{y^4} + 5) \][/tex]

Simplify further:
[tex]\[ 3x^2 z^3 \left( \frac{5y^4 + z^2}{y^4} \right) \][/tex]

So, the factored form is:
[tex]\[ 3x^2 z^3 \frac{5y^4 + z^2}{y^4} \][/tex]

### Summary of factored forms:
1. [tex]\( 10(x + y + z_3) \)[/tex]
2. [tex]\( 25xy^3 (x - 1) \)[/tex]
3. [tex]\( 3x^2 z^3 \frac{5y^4 + z^2}{y^4} \)[/tex]

These are the expressions factored step-by-step.
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