Fundamental theorem of algebra

Fundamental theorem of algebra

Basic concepts: Factoring perfect square trinomials: (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2 or (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2, Factoring trinomials, Polynomial synthetic division ,
Related concepts: Complex numbers and complex planes,

Lessons

Fundamental Theorem of Algebra:
• Every polynomial can be factored into a product of linear factors and irreducible quadratic factors.
• A degree n polynomial has exactly n roots.
  • 1.
    Introduction to Fundamental Theorem of Algebra
    For each polynomial, state the:
    • degree of the polynomial
    • nature of the roots
    • number of roots
    i) x3−2x2−13x−10x^3 - 2x^2 - 13x - 10
    ii) x3−3x2−9x−5x^3 - 3x^2 - 9x - 5
    iii) x3−6x2+8x−15x^3 - 6x^2 + 8x -15
    iv) x5−x4+x3−19x2+15x−63x^5 - x^4 + x^3 -19x^2 +15x-63
  • 2.
    Discuss the Possible Combinations of Roots
    State the possible combinations of roots for each polynomial:
    a)
    P(x)=a7x7+a6x6+a5x5+a4x4+a3x3+a2x2+a1x+a0P(x) = a_7x^7 + a_6x^6 + a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0

    b)
    P(x)=x4+...... P(x) = x^4 + ......

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3.
Polynomial Functions
3.1
What is a polynomial function?
3.2
Polynomial long division
3.3
Polynomial synthetic division
3.4
Remainder theorem
3.5
Factor theorem
3.6
Rational zero theorem
3.7
Characteristics of polynomial graphs
3.8
Multiplicities of polynomials
3.9
Imaginary zeros of polynomials
3.10
Determining the equation of a polynomial function
3.11
Applications of polynomial functions
3.12
Solving polynomial inequalities
3.13
Fundamental theorem of algebra
3.14
Descartes’ rule of signs
Grade 12 Math