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Advanced Math / Nonlinear Equations One Variable

Polynomial Equations in One Variable

Polynomial equations are often solved by moving all terms to one side, factoring, and applying the zero product property. Some SAT-style items ask for a particular solution, a nonzero solution, or the number of real solutions shown by factored form.

Least you need to know

  • If a product of factors equals zero, at least one factor must equal zero.
  • A common factor such as $x$ can create the solution $x=0$.
  • Factored form makes solutions visible.
  • When a problem asks for a positive, negative, least, greatest, or nonzero solution, give only the requested value.

Key notation

  • $x(x-5)(x+2)=0$ — A polynomial equation with solutions $0$, $5$, and $-2$
  • $x^3=16x$ — A cubic equation that can be solved by moving all terms to one side and factoring

Worked example

  • To solve $x^3=9x$, move all terms to one side: $x^3-9x=0$.
  • Factor: $x(x^2-9)=0$, so $x(x-3)(x+3)=0$.
  • The solutions are $-3$, $0$, and $3$.

Common mistakes

  • Do not divide by $x$ unless the problem has ruled out $x=0$; otherwise, you may lose a solution.
  • The degree of a polynomial does not guarantee that every solution is real.
  • When using factored form, solve each factor equal to zero.

How to recognize it

  • The problem asks for a solution to a polynomial equation.
  • The equation contains a cubic term and a common factor.
  • The problem asks for the number of real solutions from a factored equation.

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