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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], we can proceed as follows:
1. Factoring the Polynomial:
The polynomial [tex]\( f(x) = x^4 - x^3 - 39x^2 + 109x - 70 \)[/tex] can be factored into four linear factors. The factors are:
[tex]\[
f(x) = (x - 5)(x - 2)(x - 1)(x + 7)
\][/tex]
This means that when you multiply these factors together, you will get the original polynomial.
2. Solving the Equation [tex]\( f(x) = 0 \)[/tex]:
To solve [tex]\( f(x) = 0 \)[/tex], we set each factor equal to zero and solve for [tex]\( x \)[/tex]:
- [tex]\( x - 5 = 0 \)[/tex] gives [tex]\( x = 5 \)[/tex]
- [tex]\( x - 2 = 0 \)[/tex] gives [tex]\( x = 2 \)[/tex]
- [tex]\( x - 1 = 0 \)[/tex] gives [tex]\( x = 1 \)[/tex]
- [tex]\( x + 7 = 0 \)[/tex] gives [tex]\( x = -7 \)[/tex]
Therefore, the solutions to the equation [tex]\( f(x) = 0 \)[/tex] are:
[tex]\[
x = -7, \; x = 1, \; x = 2, \; x = 5
\][/tex]
In summary, the polynomial [tex]\( f(x) \)[/tex] factors as [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].
1. Factoring the Polynomial:
The polynomial [tex]\( f(x) = x^4 - x^3 - 39x^2 + 109x - 70 \)[/tex] can be factored into four linear factors. The factors are:
[tex]\[
f(x) = (x - 5)(x - 2)(x - 1)(x + 7)
\][/tex]
This means that when you multiply these factors together, you will get the original polynomial.
2. Solving the Equation [tex]\( f(x) = 0 \)[/tex]:
To solve [tex]\( f(x) = 0 \)[/tex], we set each factor equal to zero and solve for [tex]\( x \)[/tex]:
- [tex]\( x - 5 = 0 \)[/tex] gives [tex]\( x = 5 \)[/tex]
- [tex]\( x - 2 = 0 \)[/tex] gives [tex]\( x = 2 \)[/tex]
- [tex]\( x - 1 = 0 \)[/tex] gives [tex]\( x = 1 \)[/tex]
- [tex]\( x + 7 = 0 \)[/tex] gives [tex]\( x = -7 \)[/tex]
Therefore, the solutions to the equation [tex]\( f(x) = 0 \)[/tex] are:
[tex]\[
x = -7, \; x = 1, \; x = 2, \; x = 5
\][/tex]
In summary, the polynomial [tex]\( f(x) \)[/tex] factors as [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].
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