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Precalculus: Quadratics, Polynomials, and Complex Numbers

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Termini in questo insieme (21)

  • Quadratic formula
    The formula to find roots of ax² + bx + c = 0 is \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\).
  • Discriminant in quadratic formula
    The expression \(b^2-4ac\) determines the nature of roots: positive means two real roots, zero means one real root, negative means two complex roots.
  • Factoring a trinomial
    Express a quadratic \(ax^2+bx+c\) as a product of two binomials, e.g., \((mx+n)(px+q)\).
  • Steps to factor a trinomial
    Find two numbers that multiply to \(ac\) and add to \(b\), then split the middle term and factor by grouping.
  • Factoring polynomials
    Break down a polynomial into products of simpler polynomials, often starting with factoring out the greatest common factor.
  • Greatest common factor (GCF)
    The largest expression that divides all terms of a polynomial without remainder.
  • Multiplying complex numbers
    Use distributive property and apply \(i^2 = -1\) to simplify products of complex numbers.
  • Standard form of a complex number
    A complex number is written as \(a+bi\), where \(a\) and \(b\) are real numbers.
  • Factoring difference of squares
    Use the formula \(a^2 - b^2 = (a-b)(a+b)\) to factor expressions.
  • Finding solution sets of quadratic equations
    Solutions can be found by factoring, using the quadratic formula, or completing the square.
  • Completing the square
    Rewrite \(ax^2 + bx + c\) as a perfect square trinomial plus a constant to solve quadratics.
  • Multiplying binomials
    Use FOIL method: multiply First, Outer, Inner, Last terms and combine like terms.
  • Sum and product of roots
    For \(ax^2 + bx + c = 0\), sum of roots is \(-\frac{b}{a}\) and product is \(\frac{c}{a}\).
  • Polynomial degree
    The highest power of the variable in the polynomial.
  • Leading coefficient
    The coefficient of the term with the highest degree in a polynomial.
  • Zero product property
    If \(ab=0\), then either \(a=0\) or \(b=0\).
  • Conjugate of a complex number
    For \(a+bi\), the conjugate is \(a - bi\).
  • Multiplying complex conjugates
    The product of conjugates \((a+bi)(a-bi)\) equals \(a^2 + b^2\).
  • Factoring by grouping
    Group terms in pairs and factor out common factors to simplify polynomials.
  • Sum of cubes factoring formula
    Use \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\).
  • Difference of cubes factoring formula
    Use \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\).