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Precalculus Polynomial Review

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이 집합의 용어 (22)

  • Degree of polynomial p(x) = 5x(x+2)^3(x-1)
    The degree is \(5\) (1 + 3 + 1).
  • End behavior of g(x) = -2x^4 + 3x^2 - 4x + 10
    As x → ±∞, g(x) → -∞ because leading term is negative and degree is even.
  • X-intercepts of h(x) = x(x-1)^2(x+5)^3
    Zeros at x = 0, 1, and -5.
  • Other zero if polynomial has real zero at 2 and complex zero at 3 + 2i
    The other zero is 3 - 2i (complex zeros come in conjugate pairs).
  • Maximum number of real zeros for degree 8 polynomial
    Maximum of 8 real zeros.
  • Simplify (2 + 3i)(2 - 3i)
    Equals 13 (difference of squares: 2^2 - (3i)^2 = 4 + 9).
  • Simplify i^6
    Equals -1 (since i^4 = 1, i^6 = i^4 * i^2 = 1 * -1).
  • Degree of p(x) = 2x(x+2)^3(x-1)^2
    Degree is 6 (1 + 3 + 2).
  • End behavior of p(x) = 2x(x+2)^3(x-1)^2
    As x → ∞, p(x) → ∞; as x → -∞, p(x) → -∞ (leading coefficient positive, degree even).
  • Zeros, multiplicities, and behavior of p(x) = 2x(x+2)^3(x-1)^2
    Zero at 0 (mult 1, crosses), -2 (mult 3, wiggles), 1 (mult 2, bounces).
  • Maximum number of turning points for degree 6 polynomial
    Maximum turning points is 5 (degree - 1).
  • Y-intercept of p(x) = 2x(x+2)^3(x-1)^2
    Evaluate p(0) = 2*0*(2)^3*(-1)^2 = 0.
  • Form polynomial with zeros 2, 1, -1 and leading coefficient 1
    Polynomial is (x-2)(x-1)(x+1) = x^3 - 2x^2 - x + 2.
  • Is x-2 a factor of p(x) = x^3 - 2x^2 + 3x - 6?
    Use Remainder Theorem: p(2) = 0, so yes, x-2 is a factor.
  • Domain of f(x) = -2x^3 + 5x^2 - 3x + 1
    Domain is all real numbers.
  • Maximum number of real zeros for cubic polynomial
    Maximum of 3 real zeros.
  • Y-intercept of f(x) = -2x^3 + 5x^2 - 3x + 1
    Y-intercept is f(0) = 1.
  • Possible rational zeros of p(x) = x^3 + 4x^2 + x - 6
    Possible zeros are ±1, ±2, ±3, ±6.
  • Use Rational Zero Theorem to find zeros of p(x) = x^3 + 4x^2 + x - 6
    Test possible zeros to find real zeros, then factor polynomial.
  • Identify polynomial or not: f(x) = 3x^3 + 4x^2 - 1/x
    Not a polynomial (has variable in denominator).
  • Identify polynomial or not: g(x) = -200x^4 + 59x^3 - 23
    Polynomial (all terms have non-negative integer exponents).
  • Identify polynomial or not: q(x) = 3x^2 + 4x - √x
    Not a polynomial (√x = x^(1/2) is not an integer exponent).