If f(x)=x^3+8x^2-5x-84 which of the following is a factor of f(x) Answer (x-7) (x-4) (x+7) (x+3)

To determine which of the given options is a factor of \( f(x) = x^3 + 8x^2 - 5x - 84 \), we can use the Factor Theorem. According to the Factor Theorem, \( (x - c) \) is a factor of \( f(x) \) if and only if \( f(c) = 0 \).<br /><br />Let's evaluate \( f(x) \) at the roots corresponding to each factor:<br /><br />1. For \( (x-7) \), evaluate \( f(7) \):<br /> \[<br /> f(7) = 7^3 + 8 \times 7^2 - 5 \times 7 - 84<br /> \]<br /> \[<br /> = 343 + 8 \times 49 - 35 - 84<br /> \]<br /> \[<br /> = 343 + 392 - 35 - 84<br /> \]<br /> \[<br /> = 700 - 119<br /> \]<br /> \[<br /> = 581<br /> \]<br /> Since \( f(7) \neq 0 \), \( (x-7) \) is not a factor.<br /><br />2. For \( (x-4) \), evaluate \( f(4) \):<br /> \[<br /> f(4) = 4^3 + 8 \times 4^2 - 5 \times 4 - 84<br /> \]<br /> \[<br /> = 64 + 8 \times 16 - 20 - 84<br /> \]<br /> \[<br /> = 64 + 128 - 20 - 84<br /> \]<br /> \[<br /> = 192 - 104<br /> \]<br /> \[<br /> = 88<br /> \]<br /> Since \( f(4) \neq 0 \), \( (x-4) \) is not a factor.<br /><br />3. For \( (x+7) \), evaluate \( f(-7) \):<br /> \[<br /> f(-7) = (-7)^3 + 8 \times (-7)^2 - 5 \times (-7) - 84<br /> \]<br /> \[<br /> = -343 + 8 \times 49 + 35 - 84<br /> \]<br /> \[<br /> = -343 + 392 + 35 - 84<br /> \]<br /> \[<br /> = 427 - 427<br /> \]<br /> \[<br /> = 0<br /> \]<br /> Since \( f(-7) = 0 \), \( (x+7) \) is a factor.<br /><br />4. For \( (x+3) \), evaluate \( f(-3) \):<br /> \[<br /> f(-3) = (-3)^3 + 8 \times (-3)^2 - 5 \times (-3) - 84<br /> \]<br /> \[<br /> = -27 + 8 \times 9 + 15 - 84<br /> \]<br /> \[<br /> = -27 + 72 + 15 - 84<br /> \]<br /> \[<br /> = 60 - 111<br /> \]<br /> \[<br /> = -51<br /> \]<br /> Since \( f(-3) \neq 0 \), \( (x+3) \) is not a factor.<br /><br />Therefore, the correct answer is \( (x+7) \).