White Plains High School

PCH 6.1 Intro to POLYNOMIALSMr. Stanton

SWBAT factor a polynomial function

SWBAT write the equation of a polynomial given its roots

A polynomial in the FIELD[1] OF REAL NUMBERS is an expression of the form

where

  1. Based on the definition, which ones of the following are polynomials in the field of real numbers?
  2. F(x) = 2x + 1
  3. G(x) = 5
  4. L(x) = 2x5 – 4x + 8
  1. The ______of a polynomial (in one variable) is the value of the highest exponent.
  1. A polynomial is written in standard form when written in order of ______exponents.
  1. The ______term of the polynomial is the term that is the coefficient of the variable raised to the zero power.
  1. The ______coefficient of the polynomial is the coefficient of the variable with the highest exponent.
  1. Where do the words “quotient,” “divisor,” and “dividend” go in this pretty little division symbol?

Pre-Calc Honors: Roots, Factors, and Polynomials

Factor each of the following completely over the field of Rational Numbers:

a) X2 – 9 / b) x2 – 12 / c) X2 + 9 / d) x4 + 3x2 - 4

Factor each of the following completely over the field of Real Numbers:

e) X2 – 9 / f) x2 – 12 / g) X2 + 9 / h) x4 + 3x2 - 4

Factor each of the following completely over the field of Complex Numbers:

i) X2 – 9 / j) x2 – 12 / k) X2 + 9 / l) x4 + 3x2 - 4

Vocabulary (p. 201 – 202) in simple English:

  • Depressed Equation: The resulting polynomial when P(x) is divided by factor (x – c)
  • Multiplicity of a root: Let’s say the complete factorization of P(x) = (x – c)3(x – d), then the root x = c has a multiplicity 3.
  • Reducible polynomial over a field: P(x) is a reducible polynomial if it is further factorable over that field.
  • Irreducible polynomial over a field: P(x) is an irreducible polynomial if it is not further factorable over that field.

Relationship between roots, polynomial equations, and factors

  1. If x = -4 is a root of polynomial P(x), then a factor of P(x) must be ______.
  2. If is a root of polynomial P(x), then P(x) must have a factor of ______.
  3. If is a root of polynomial P(x), then ______must also be a root of P(x).
  4. How many roots of polynomial P(x) are there if P(x) has degree 5?______.
  5. Based on the previous 2 questions: How many IMAGINARY roots of polynomial P(x) are there if P(x) has degree 5? ______
  6. Recall that at some point during this year I asked you to write a quadratic with roots and , or something like that. Does this contradict what you wrote in #4? Why or why not?

A2T Review: Determining a Polynomial Given the Roots

If the roots of a polynomial are r1, r2, r3...rn then P(x) can be represented in standard form by multiplying out (x – r1)(x – r2)(x – r3)…(x – rn).

*If the polynomial needs to be written in the field of rational numbers, then you need to multiply it by some constant c that eliminates all of the coefficients that are not rational.

Write a quadratic equation that has:

a) Roots

b) Roots

c) Roots

d) Roots passing through the point (6, -10)

e) Vertex (-2, 6)

f) Vertex (-2, 6) passing through the point (2, -18)

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[1] Please read p. 196 of Dolciani to get a better idea of what a FIELD is. The set of real numbers is a FIELD. The set of complex number is a FIELD. The set of rational numbers is a FIELD.