High School

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Factor the expression:

[tex]\[ 15x^4 - 30x^3 + 45x^2 \][/tex]

Answer :

We start with the polynomial

[tex]$$15x^4 - 30x^3 + 45x^2.$$[/tex]

### Step 1: Identify the Greatest Common Factor (GCF)

Notice that each term in the polynomial has a factor of [tex]$15$[/tex] and at least [tex]$x^2$[/tex]. Therefore, the GCF is

[tex]$$15x^2.$$[/tex]

### Step 2: Factor Out the GCF

We factor [tex]$15x^2$[/tex] out of each term:

- For [tex]$15x^4$[/tex]:
[tex]$$15x^4 = 15x^2 \cdot x^2.$$[/tex]

- For [tex]$-30x^3$[/tex]:
[tex]$$-30x^3 = 15x^2 \cdot (-2x).$$[/tex]

- For [tex]$45x^2$[/tex]:
[tex]$$45x^2 = 15x^2 \cdot 3.$$[/tex]

Thus, the polynomial can be rewritten as:

[tex]$$15x^4 - 30x^3 + 45x^2 = 15x^2(x^2 - 2x + 3).$$[/tex]

### Step 3: Write the Final Factored Form

The completely factored form of the polynomial is

[tex]$$15x^2(x^2 - 2x + 3).$$[/tex]

This is the final answer.

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