Answer :
To factor the expression [tex]\(-5x^3 + 45x^2 - 90x\)[/tex] completely, we'll start by identifying the greatest common factor (GCF) of all the terms.
### Step 1: Find the GCF
1. Look at the coefficients: [tex]\(-5\)[/tex], [tex]\(45\)[/tex], and [tex]\(-90\)[/tex].
- The GCF of [tex]\(-5\)[/tex], [tex]\(45\)[/tex], and [tex]\(-90\)[/tex] is [tex]\(5\)[/tex].
2. All the terms have a factor of [tex]\(x\)[/tex] as well. The lowest power of [tex]\(x\)[/tex] in the terms is [tex]\(x\)[/tex].
So, the GCF of all the terms is [tex]\(5x\)[/tex].
### Step 2: Factor out the GCF
1. Factor [tex]\(5x\)[/tex] out of each term:
[tex]\[
-5x^3 + 45x^2 - 90x = 5x(-x^2 + 9x - 18)
\][/tex]
### Step 3: Factor the quadratic expression
Now, we focus on factoring the quadratic [tex]\(-x^2 + 9x - 18\)[/tex].
1. Notice that the quadratic has a leading coefficient of [tex]\(-1\)[/tex]. First, factor out [tex]\(-1\)[/tex]:
[tex]\[
-x^2 + 9x - 18 = -(x^2 - 9x + 18)
\][/tex]
2. Factor the quadratic expression [tex]\(x^2 - 9x + 18\)[/tex]:
- We need two numbers that multiply to [tex]\(18\)[/tex] (the constant term) and add to [tex]\(-9\)[/tex] (the coefficient of [tex]\(x\)[/tex].
- The numbers [tex]\(-3\)[/tex] and [tex]\(-6\)[/tex] satisfy these conditions because [tex]\(-3 \times -6 = 18\)[/tex] and [tex]\(-3 + (-6) = -9\)[/tex].
3. Write the factored form:
[tex]\[
x^2 - 9x + 18 = (x - 3)(x - 6)
\][/tex]
### Step 4: Put it all together
Now substitute back, including the negative sign factored out earlier:
[tex]\[
-(x^2 - 9x + 18) = -(x - 3)(x - 6)
\][/tex]
Therefore, the completely factored form of [tex]\(-5x^3 + 45x^2 - 90x\)[/tex] is
[tex]\[
5x(-(x - 3)(x - 6))
\][/tex]
Simplify the expression to present it more neatly:
[tex]\[
-5x(x - 3)(x - 6)
\][/tex]
And there you have it, the expression [tex]\(-5x^3 + 45x^2 - 90x\)[/tex] fully factored is [tex]\(-5x(x - 3)(x - 6)\)[/tex].
### Step 1: Find the GCF
1. Look at the coefficients: [tex]\(-5\)[/tex], [tex]\(45\)[/tex], and [tex]\(-90\)[/tex].
- The GCF of [tex]\(-5\)[/tex], [tex]\(45\)[/tex], and [tex]\(-90\)[/tex] is [tex]\(5\)[/tex].
2. All the terms have a factor of [tex]\(x\)[/tex] as well. The lowest power of [tex]\(x\)[/tex] in the terms is [tex]\(x\)[/tex].
So, the GCF of all the terms is [tex]\(5x\)[/tex].
### Step 2: Factor out the GCF
1. Factor [tex]\(5x\)[/tex] out of each term:
[tex]\[
-5x^3 + 45x^2 - 90x = 5x(-x^2 + 9x - 18)
\][/tex]
### Step 3: Factor the quadratic expression
Now, we focus on factoring the quadratic [tex]\(-x^2 + 9x - 18\)[/tex].
1. Notice that the quadratic has a leading coefficient of [tex]\(-1\)[/tex]. First, factor out [tex]\(-1\)[/tex]:
[tex]\[
-x^2 + 9x - 18 = -(x^2 - 9x + 18)
\][/tex]
2. Factor the quadratic expression [tex]\(x^2 - 9x + 18\)[/tex]:
- We need two numbers that multiply to [tex]\(18\)[/tex] (the constant term) and add to [tex]\(-9\)[/tex] (the coefficient of [tex]\(x\)[/tex].
- The numbers [tex]\(-3\)[/tex] and [tex]\(-6\)[/tex] satisfy these conditions because [tex]\(-3 \times -6 = 18\)[/tex] and [tex]\(-3 + (-6) = -9\)[/tex].
3. Write the factored form:
[tex]\[
x^2 - 9x + 18 = (x - 3)(x - 6)
\][/tex]
### Step 4: Put it all together
Now substitute back, including the negative sign factored out earlier:
[tex]\[
-(x^2 - 9x + 18) = -(x - 3)(x - 6)
\][/tex]
Therefore, the completely factored form of [tex]\(-5x^3 + 45x^2 - 90x\)[/tex] is
[tex]\[
5x(-(x - 3)(x - 6))
\][/tex]
Simplify the expression to present it more neatly:
[tex]\[
-5x(x - 3)(x - 6)
\][/tex]
And there you have it, the expression [tex]\(-5x^3 + 45x^2 - 90x\)[/tex] fully factored is [tex]\(-5x(x - 3)(x - 6)\)[/tex].
