Answer :
To factor the polynomial [tex]\( f(x) = x^4 - x^3 - 39x^2 + 109x - 70 \)[/tex] and solve the equation [tex]\( f(x) = 0 \)[/tex], follow these steps:
### Step 1: Factor the Polynomial
The polynomial can be factored into simpler expressions, specifically into its linear factors. The given polynomial factors into:
[tex]\[ f(x) = (x - 5)(x - 2)(x - 1)(x + 7) \][/tex]
This means that the polynomial can be expressed as a product of these linear terms.
### Step 2: Solve the Equation [tex]\( f(x) = 0 \)[/tex]
To solve the equation [tex]\( f(x) = 0 \)[/tex], we set each factor equal to zero and solve for [tex]\( x \)[/tex]:
1. [tex]\( x - 5 = 0 \)[/tex]
[tex]\[ x = 5 \][/tex]
2. [tex]\( x - 2 = 0 \)[/tex]
[tex]\[ x = 2 \][/tex]
3. [tex]\( x - 1 = 0 \)[/tex]
[tex]\[ x = 1 \][/tex]
4. [tex]\( x + 7 = 0 \)[/tex]
[tex]\[ x = -7 \][/tex]
These steps show us that the solutions to the equation [tex]\( f(x) = 0 \)[/tex] are [tex]\( x = -7, 1, 2, \)[/tex] and [tex]\( 5 \)[/tex].
### Final Answer
So, the polynomial [tex]\( f(x) = x^4 - x^3 - 39x^2 + 109x - 70 \)[/tex] factors into [tex]\((x - 5)(x - 2)(x - 1)(x + 7)\)[/tex], and the solutions to the equation [tex]\( f(x) = 0 \)[/tex] are [tex]\( x = -7, 1, 2, \)[/tex] and [tex]\( 5 \)[/tex].
### Step 1: Factor the Polynomial
The polynomial can be factored into simpler expressions, specifically into its linear factors. The given polynomial factors into:
[tex]\[ f(x) = (x - 5)(x - 2)(x - 1)(x + 7) \][/tex]
This means that the polynomial can be expressed as a product of these linear terms.
### Step 2: Solve the Equation [tex]\( f(x) = 0 \)[/tex]
To solve the equation [tex]\( f(x) = 0 \)[/tex], we set each factor equal to zero and solve for [tex]\( x \)[/tex]:
1. [tex]\( x - 5 = 0 \)[/tex]
[tex]\[ x = 5 \][/tex]
2. [tex]\( x - 2 = 0 \)[/tex]
[tex]\[ x = 2 \][/tex]
3. [tex]\( x - 1 = 0 \)[/tex]
[tex]\[ x = 1 \][/tex]
4. [tex]\( x + 7 = 0 \)[/tex]
[tex]\[ x = -7 \][/tex]
These steps show us that the solutions to the equation [tex]\( f(x) = 0 \)[/tex] are [tex]\( x = -7, 1, 2, \)[/tex] and [tex]\( 5 \)[/tex].
### Final Answer
So, the polynomial [tex]\( f(x) = x^4 - x^3 - 39x^2 + 109x - 70 \)[/tex] factors into [tex]\((x - 5)(x - 2)(x - 1)(x + 7)\)[/tex], and the solutions to the equation [tex]\( f(x) = 0 \)[/tex] are [tex]\( x = -7, 1, 2, \)[/tex] and [tex]\( 5 \)[/tex].
