Factor completely.
3, x, cubed, minus, 5, x, squared, minus, 27, x, plus, 45
3x^3−5x^2−27x+45
To factor completely, we can first use the Rational Root Theorem to find any possible rational roots.
The Rational Root Theorem states that if a polynomial:
p(x) = a_n * x^n + a_n-1 * x^(n-1) + ... + a_1 * x + a_0
has a rational root x = p/q, where p is a factor of the constant term a_0 and q is a factor of the leading coefficient a_n.
In our case, the constant term is 45, and the leading coefficient is 3.
The factors of 45 are ±1, ±3, ±5, ±9, ±15, ±45.
The factors of 3 are ±1, ±3.
Using the Rational Root Theorem, the possible rational roots are:
±1, ±3, ±5, ±9, ±15, ±45(possible numerators) /
±1, ±3(possible denominators)
Let's try some of these possibilities by substituting them into the polynomial:
For x = 1:
3(1)^3 - 5(1)^2 - 27(1) + 45 = 3 - 5 - 27 + 45 = 16, which is not equal to 0.
For x = -1:
3(-1)^3 - 5(-1)^2 - 27(-1) + 45 = -3 - 5 + 27 + 45 = 64, which is not equal to 0.
For x = 3:
3(3)^3 - 5(3)^2 - 27(3) + 45 = 81 - 45 - 81 + 45 = 0.
Therefore, x = 3 is a root.
Using synthetic division, we can divide the polynomial by x - 3:
```
3 | 3 -5 -27 45
| 9 12 -45
+____________________
3 4 -15 0
```
The quotient is 3x^2 + 4x - 15, and the remainder is 0.
So, we have factored the polynomial as (x - 3)(3x^2 + 4x - 15).
Next, we can factor the quadratic expression 3x^2 + 4x - 15.
We look for two numbers whose product is -45 (the product of the coefficient of x^2 and the constant term), and whose sum is 4 (the coefficient of x).
The numbers that fit these conditions are 9 and -5.
So, we can factor 3x^2 + 4x - 15 as (x + 3)(3x - 5).
Therefore, the completely factored form of the polynomial 3x^3 - 5x^2 - 27x + 45 is:
(x - 3)(x + 3)(3x - 5).
To factor the expression 3x^3 - 5x^2 - 27x + 45 completely, we can start by looking for common factors. From the given expression, we can see that each term is divisible by 3. So, we can factor out 3 as a common factor:
3(x^3 - (5/3)x^2 - 9x + 15)
Now we need to factor the quadratic expression inside the parentheses. To do that, we can look for factors of the leading coefficient (1) and the constant term (15) that add up to the coefficient of the middle term (-5/3). In this case, the factors of 1 and 15 that satisfy these conditions are 3 and -5:
3(x^3 - 5/3x^2 - 9x + 15)
= 3(x^2(x - 5/3) - 3(3x - 5))
Now we can factor the first two terms in the parentheses by using the distributive property:
3(x - 5/3)(x^2 - 3(3x - 5))
Simplifying further:
3(x - 5/3)(x^2 - 9x + 15)
Therefore, the completely factored expression is 3(x - 5/3)(x^2 - 9x + 15).
To factor the given expression completely, we can use a method called factoring by grouping. Here's how you can do it:
Step 1: Group the terms in pairs.
(3x^3 - 5x^2) - (27x - 45)
Step 2: Factor out the greatest common factor from each group.
x^2(3x - 5) - 9(3x - 5)
Step 3: Notice that both groups have a common binomial factor of (3x - 5). Factor it out.
(3x - 5)(x^2 - 9)
Step 4: Further factor the second term using the difference of squares formula.
(3x - 5)(x - 3)(x + 3)
Therefore, the completely factored form of the expression 3x^3 - 5x^2 - 27x + 45 is (3x - 5)(x - 3)(x + 3).