which statements are true about the polynomial function?\n$f(x)=x^{4}+5x^{3}-x^{2}-5x$\n$\\square…

which statements are true about the polynomial function?\n$f(x)=x^{4}+5x^{3}-x^{2}-5x$\n$\\square f(5)=0$\n$\\square f(x)=0$ when $x=-5$\n$\\square f(x)$ divided by $(x + 5)$ has a remainder of 0.\n$\\square (x - 5)$ is a factor of $f(x)$.

which statements are true about the polynomial function?\n$f(x)=x^{4}+5x^{3}-x^{2}-5x$\n$\\square f(5)=0$\n$\\square f(x)=0$ when $x=-5$\n$\\square f(x)$ divided by $(x + 5)$ has a remainder of 0.\n$\\square (x - 5)$ is a factor of $f(x)$.

Answer

Explanation:

Step1: Factor the polynomial function

Factor (f(x)=x^{4}+5x^{3}-x^{2}-5x) by grouping: [ \begin{align*} f(x)&=x(x^{3}+5x^{2}-x - 5)\ &=x\left[x^{2}(x + 5)-(x + 5)\right]\ &=x(x + 5)(x^{2}-1)\ &=x(x + 5)(x - 1)(x + 1) \end{align*} ]

Step2: Check each statement

  • For (f(5)): Substitute (x = 5) into (f(x)): (f(5)=5\times(5 + 5)\times(5 - 1)\times(5 + 1)=5\times10\times4\times6=1200\neq0)
  • For (x=-5): Substitute (x=-5) into (f(x)): (f(-5)=-5\times(-5 + 5)\times(-5 - 1)\times(-5 + 1)=-5\times0\times(-6)\times(-4)=0)
  • For division by ((x + 5)): Since (f(x)) has a factor of ((x + 5)), by the factor theorem, (f(x)) divided by ((x + 5)) has a remainder of (0)
  • For ((x - 5)) as a factor: Since (f(5)\neq0), ((x - 5)) is not a factor of (f(x))

Answer:

(f(x)=0) when (x = - 5) and (f(x)) divided by ((x + 5)) has a remainder of (0)