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ax3+bx2+cx+d=0ax^{3}+bx^{2}+cx+d=0\newlineIn the equation above, \newlinea,b,c,a,b,c, and \newlinedd are constants.\newlineIf the equation has roots \newline1,3,-1,-3, and 55, which of the following is a factor of \newlineax3+bx2+cx+dax^{3}+bx^{2}+cx+d ?\newlineA) x1x-1\newlineB) x+1x+1\newlineC) x3x-3\newlineD) x+5x+5

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Q. ax3+bx2+cx+d=0ax^{3}+bx^{2}+cx+d=0\newlineIn the equation above, \newlinea,b,c,a,b,c, and \newlinedd are constants.\newlineIf the equation has roots \newline1,3,-1,-3, and 55, which of the following is a factor of \newlineax3+bx2+cx+dax^{3}+bx^{2}+cx+d ?\newlineA) x1x-1\newlineB) x+1x+1\newlineC) x3x-3\newlineD) x+5x+5
  1. Identify Relationship: Identify the relationship between roots and factors of a polynomial.\newlineIf a polynomial has roots at x=rx = r, then (xr)(x - r) is a factor of the polynomial.
  2. Determine Factors: Given the roots 1-1, 3-3, and 55, determine the corresponding factors.\newlineThe factors corresponding to these roots are (x(1))(x - (-1)), (x(3))(x - (-3)), and (x5)(x - 5), which simplify to (x+1)(x + 1), (x+3)(x + 3), and (x5)(x - 5) respectively.
  3. Match with Choices: Match the given factors with the choices provided.\newlineThe factors we found are (x+1)(x + 1), (x+3)(x + 3), and (x5)(x - 5). We need to find which one of these is listed in the choices.
  4. Identify Correct Factor: Identify the correct factor from the choices.\newlineChoice A is (x1)(x - 1), which does not match any of our factors.\newlineChoice B is (x+1)(x + 1), which matches one of our factors.\newlineChoice C is (x3)(x - 3), which does not match any of our factors.\newlineChoice D is (x+5)(x + 5), which does not match any of our factors.

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