Factor completely.
3, x, cubed, minus, 5, x, squared, minus, 27, x, plus, 45
3x
3
−5x
2
−27x+45
The completely factored form of the expression is:
(3x - 3)(x^2 - 5x - 9)
To factor completely, let's start by looking for common factors of all the terms.
First, let's look for common factors of the coefficients: 3, -5, -27, and 45. The only common factor is 1.
Next, let's look for common factors of the variables: x. The variables cannot be factored further.
Now, let's factor each term using the distributive property:
3x^3 - 5x^2 - 27x + 45
= 3x(x^2) - 5(x^2) - 27x + 45
= x(x^2 - 5) - 9(3x - 5)
= x(x^2 - 5) - 9(3x - 5)
We can see that we have a common factor of (x^2 - 5) in the first two terms and a common factor of (3x - 5) in the last two terms.
Factoring this out, we get:
= (x - sqrt(5))(x + sqrt(5)) - 9(3x - 5)
= (x - sqrt(5))(x + sqrt(5)) - 27x + 45
So the completely factored form of the expression 3x^3 - 5x^2 - 27x + 45 is:
(x - sqrt(5))(x + sqrt(5)) - 27x + 45
To factor the expression completely, we need to look for common factors among the terms. In this case, the terms do not have any common factors, so we will use a different method called factoring by grouping.
The expression we have is:
3x^3 - 5x^2 - 27x + 45
Step 1: Group the terms in pairs:
(3x^3 - 5x^2) + (-27x + 45)
Step 2: Factor out the greatest common factor (GCF) from each pair separately:
GCF of the first pair: x^2
GCF of the second pair: -9
Rewriting the expression:
x^2(3x - 5) - 9(3x - 5)
Step 3: Notice that we now have a common binomial factor, (3x - 5). We can factor it out:
(3x - 5)(x^2 - 9)
Step 4: Further factor the difference of squares, x^2 - 9:
(3x - 5)(x - 3)(x + 3)
So, the expression 3x^3 - 5x^2 - 27x + 45 completely factors as (3x - 5)(x - 3)(x + 3).