Chapter 4 Vocabulary
1. Polynomial in one variable – an expression of the form 𝑎0 𝑥 𝑛 + 𝑎1 𝑥 𝑛−1 + ⋯ + 𝑎𝑛−2 𝑥 2 + 𝑎𝑛−1 𝑥 +
𝑎𝑛 . The coefficients represent complex numbers (real or imaginary) 𝑎0 is not zero, and n represents a
nonnegative integer.
2. Degree – the greatest exponent of a polynomials variable.
3. Leading coefficient – the coefficient of the variable with the greatest exponent.
4. Polynomial function – a function 𝑦 = 𝑃(𝑥) where P(x) is a polynomial in one variable.
5. Polynomial equation - a polynomial that is set equal to zero.
6. Root – a solution of the equation P(x) = 0
7. Imaginary Number – a complex number of the form 𝑎 = 𝑏𝑖 where 𝑏 ≠ 0 and 𝑖 is the imaginary
number.
8. Complex numbers – the imaginary numbers combined with the real numbers
9. Pure Imaginary Number – the complex number 𝑎 + 𝑏𝑖 when a = 0 and 𝑏 ≠ 0
10. Fundamental Theorem of Algebra – Every polynomial equation with degree greater than zero has at
least one root in the set of complex numbers.
11. Completing the square – a process used to create a perfect square trinomial.
12. Quadratic Formula –
𝑥=
−𝑏±√𝑏2 −4𝑎𝑐
2𝑎
13. Discriminant – the radicand of the quadratic formula 𝑏 2 − 4𝑎𝑐
14. Conjugates – A pair of complex numbers 𝑎 + 𝑏𝑖 𝑎𝑛𝑑 𝑎 − 𝑏𝑖
15. Complex Conjugates Theorem – Suppose a and b are real numbers with b≠0, if 𝑎 + 𝑏𝑖 is a root of a
polynomial equation with real coefficients, then a−𝑏𝑖 is also a root ot the equation.
16. Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant, P(r), and
𝑃(𝑥) = (𝑥 − 𝑟) ∙ 𝑄(𝑥) + 𝑃(𝑟), where Q(x) is a polynomial with degree one less than the degree of
P(x).
17. Synthetic Division – shortcut for dividing a polynomial by a binomial of the form x-r.
18. Factor Theorem – The binomial x-r is a factor of the polynomial P(x) if and only if P(r) = 0
19. Rational Root Theorem - Let 𝑎0 𝑥 𝑛 + 𝑎1 𝑥 𝑛−1 + ⋯ + 𝑎𝑛−1 𝑥 + 𝑎𝑛 = 0 represent a polynomial
𝑝
equation of degree n with integral coefficients. If a rational number 𝑞 , where p and q have no
20.
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common factors, is a root of the equation, then p is a factor of 𝑎𝑛 and q is a factor of 𝑎0
Integral Root Theorem - - Let 𝑎0 𝑥 𝑛 + 𝑎1 𝑥 𝑛−1 + ⋯ + 𝑎𝑛−1 𝑥 + 𝑎𝑛 = 0 represent a polynomial
equation that has a leading coefficient of 1, integral coefficients, and 𝑎𝑛 ≠ 0. Any rational roots of
this equation must be integral factors of 𝑎𝑛 .
Descartes’ Rule of Signs – can be used to determine the possible number of positive real zeros a
polynomial has.
Location Principle – Suppose 𝑦 = 𝑓(𝑥) represents a polynomial function with real coefficients. If a
and b are two numbers with f(a) and f(b) positive, the function has at least one real zero between a
and b.
Upper Bound – an integer greater than or equal to the greatest real zero.
Lower Bound – an integer less than or equal to the least real zero.
Upper Bound Theorem – helps to confirm whether you have determined all of the real zeros.