To factor the polynomial [tex]\( f(x) = 3x^3 + 5x^2 - 26x + 8 \)[/tex] completely given that -4 is a root, we follow these steps:
### Step 1: Verify the Given Root
First, we verify that -4 is indeed a root of the polynomial:
[tex]\[ f(-4) = 3(-4)^3 + 5(-4)^2 - 26(-4) + 8 \][/tex]
[tex]\[ f(-4) = 3(-64) + 5(16) + 104 + 8 \][/tex]
[tex]\[ f(-4) = -192 + 80 + 104 + 8 \][/tex]
[tex]\[ f(-4) = 0 \][/tex]
Since [tex]\( f(-4) = 0 \)[/tex], -4 is indeed a root of [tex]\( f(x) \)[/tex].
### Step 2: Polynomial Division
Now, we perform polynomial division of [tex]\( f(x) \)[/tex] by [tex]\( x + 4 \)[/tex]. This will give us a quotient and a remainder:
[tex]\[ \text{Divide } f(x) \text{ by } x + 4 \][/tex]
Upon dividing, we find:
[tex]\[ f(x) = (x + 4)(3x^2 - 7x + 2) \][/tex]
### Step 3: Factor the Quadratic Polynomial
Next, we need to factor the quadratic polynomial [tex]\( 3x^2 - 7x + 2 \)[/tex]. We look for factors of [tex]\( 3 \cdot 2 = 6 \)[/tex] that add up to -7. The factors -1 and -6 satisfy this condition. So, we can rewrite and factor the quadratic expression as:
[tex]\[ 3x^2 - 7x + 2 \][/tex]
[tex]\[ = 3x^2 - x - 6x + 2 \][/tex]
[tex]\[ = x(3x - 1) - 2(3x - 1) \][/tex]
[tex]\[ = (3x - 1)(x - 2) \][/tex]
### Step 4: Combine the Factors
Combining all the factors, we get:
[tex]\[ f(x) = (x + 4)(3x - 1)(x - 2) \][/tex]
So, the completely factored form of the polynomial [tex]\( 3x^3 + 5x^2 - 26x + 8 \)[/tex] is:
[tex]\[ f(x) = (x + 4)(3x - 1)(x - 2) \][/tex]
Thus, the polynomial [tex]\( f(x) = 3 x^3 + 5 x^2 - 26 x + 8 \)[/tex] can be factored completely as:
[tex]\[ f(x) = (x + 4)(3x - 1)(x - 2) \][/tex]
