Fundamental Theorem of Algebra?

Find the zeros of the function by factoring and using the Zero Product Property.

a. f(x) = x^3 +9x

b. g(x) = (x - 2)^2 + 4(x - 2) + 4

I'm working on my midterm review and am stuck on this. I really can't figure it out. A step-by-step guide on how to solve this would be excruciatingly helpful. Thank you in advance; it's greatly appreciated.

Comments

  • Find the zeros of the function by factoring and using the Zero Product Property.

    a. f(x) = x^3 +9x

    0= x^3+9x

    0= x(x^2+9)

    The zero product property says that if a product = 0, then one of the factors is zero.

    So x= 0 or x^2+ 9= 0

    X= 0 or x^2= -9

    The second equation has no real solution, so it would just be x= 0

    but if you also want complex solutions, then x= 0 or +/- 3i

    b. g(x) = (x - 2)^2 + 4(x - 2) + 4

    You could just expand and simplify the function, but I would write this by substitution. Let u = x-2

    0= u^2 +4u +4

    0 = (u+2)^2

    u= -2

    Then x-2= -2

    So x= 0

    Be sure to always check the answers.

    Hoping this helps!

  • The zero product property just means that anything times 0 = 0 and if you have a product of 0 then you multiplied something by zero

    x^3 +9x

    x(x^2 + 9) = 0

    Since the product is zero, we no that either x or (x2 + 9) needs to be zero for this to be true

    x^2 + 9 = 0

    x^2 = -9

    sqrt of a negative gives an imaginary number (3i in this case) but we still know x = 0 is a solution.

  • x^3 + 9x

    = x( x^2 + 9) =0

    so x= 0 & x = 3 i answer

    =====================

    (x-2)^2 + 4 (x-2) + 4 =0

    { ( x-2) + 2 }^2 =0

    or ( x-2 ) + 2 =0

    or x = 0 answer

  • you have roots of an equation and zeroes of a polynomial. F(x)/x^4=a + b/x+c/x^2+d/x^3+e/x^4 and if X=a million/x you get eX^4+dX^3+cX^2+bX+a so the roots of g(x)=0 are the reciprocals of the roots of f(x)=0.

  • a. x(x + 3i)(x – 3i) = 0, x = 0, ±3i

    b. ((x – 2) + 2)² = x² = 0, x = 0

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