Algebra 2 Honors - Unit 3 (Polynomials)

3.1 - Polynomials

Polynomial in x: any expression of the form $a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots+a_1x+a_0$

Where n is a nonnegative integer and the coefficients a₀…a_n are real numbers

Let $f\left(x\right)=2x^4+x^2-5$ :

Coefficients are 2, 1, and -5
The degree of each term is the power of the variable
So $2x^4$ has degree 4
And the degree of the polynomial is 4, the highest/largest power

Long-term Behavior/Concavity/Turning Points

1st Degree = line (Linear)
1. $P\left(x\right)=ax+b$
2nd Degree = Parabola (Quadratic)
1. $P\left(x\right)=ax^2+bx+c$
3rd Degree = Cubic
1. $P\left(x\right)=ax^3+bx^2+cx+d$
4th Degree = Quartic
1. $P\left(x\right)=ax^4+bx^3+\cdots+e$

KNOW THE GRAPH SHAPES

Turning points: function values change from increasing to decreasing, or vice versa

Relative minimum: changes from increasing to decreasing

Relative maximum: changes from decreasing to increasing

Both relatives are local

End Behavior:

As x or y goes to infinity, how does x or y respond?

3.3 - Polynomial Identities

To add or subtract polynomials, collect the like terms and add/subtract the coefficients

To multiply 2 polynomials: Multiply every term of the first to every term of the second, then combine like terms.

Additive Inverse: Two values that added will equal 0.

Ex: the additive inverse of $5x^3-7x^2+3x-6$ is $-5x^3+7x^2-3x+6$

CUBES:

$A^3+B^3=\left(A+B\right)\left(A^2-ab+B^2\right)$

$A^3-B^3=\left(A-B\right)\left(A^2+ab+B^2\right)$

KNOW PASCAL’S TRIANGLE

3.4 - Division of Polynomials/Remainder Theorem/Zeros

HERE WE LEARN SYNTHETIC DIVISION, SO REVIEW IT

Root: a solution to a polynomial equation P(x) = 0

Zero: a number that makes a polynomial function P(x) zero

Factor: if you divide a possible value by the larger equation and there’s no remainder, that value is a factor.

Quotient-Remainder Theorem: if P(x) is a polynomial and D is not equal to 0, then there are polynomials Q(x) and R(x) such that:

$P\left(x\right)=D\left(x\right)\cdot Q\left(x\right)+R\left(x\right)$ or Dividend = (Divisor)(quotient) + remainder

where R(x) = 0 or R(x) < degree D(x)

The Remainder Theorem: P(x) = The remainder of dividing the equation by x

EX: If $P\left(x\right)=3x^7+10x^6+33x^4-15x^2+6x-9$ , then find P(-4)

Here divide by -4, and the remainder is the answer.

The Factor Theorem: for a polynomial P(x), if P(r) = 0. then

r is a ROOT of P(x) = 0
x-r is a factor of P(x)

REVIEW

3.4c

$\frac{f\left(x\right)}{g\left(x\right)}=q\left(x\right)+\frac{r\left(x\right)}{g\left(x\right)}$ or $\left.f\left(x\right.\right)=q\left(x\right)g\left(x\right)+r\left(x\right)$

With the factor theorem":

if f(c) = 0, then x-c is a factor of f(x).
if x-c is a factor of f(x), then f(c) = 0.

PRACTICE THESE QUESTIONS

3.5 - Zeroes of a Polynomial Function

REVIEW THROUGH PAPER IT TALKS ABOUT GRAPHING THESE FUNCTIONS

3.6 - Theorems About Roots of Polynomials

Fundamental Theorem of Algebra:
1. Every polynomial of degree n > 1 with complex (ANY) coefficients, has at least one linear factor: x - a
2. Every polynomial of degree n > 1, with complex (ANY) coefficients, can be factored into EXACTLY n linear factors
3. Ex: How many linear factors does the polynomial $P\left(x\right)=\left(x^2-5x-14\right)^3$ have?
  1. Degree 6 polynomial, so must have 6 zeroes or 6 factors
  2. MULTIPLICITY: the number of times a factor occurs
Conjugate Root Theorems
1. Rational Coefficients: If $a+c\sqrt{b}$ is a root of a polynomial with RATIONAL coefficients, then $a-c\sqrt{b}$ is also a root
2. Real Coefficients: If a + bi is a root of a polynomial with REAL coefficients, then a - bi is also a root.
Rational Roots Theorem
1. Consider a rational number c/d where c and d have no common factors.
2. If c/d is a root of P(x), c must be a factor of a₀ and d must be a factor of a_n
3. Ex: List the possible RATIONAL roots of $P\left(x\right)=2x^3+x^2+x-6$
  1. a_n = 2, and a₀ = -6
  2. So c = $\pm1,\pm2,\pm3,\pm6$ (all factors of -6)
  3. So d = $\pm1,\pm2$ (all factors of 2)
4. IF COEFFICIENTS ARE ALL POSITIVE, THERE ARE NO POSITIVE ROOTS!
Descartes’ Rule of Signs
1. Positive Roots
  1. The number of positive REAL roots (+RR) of P(x) with REAL coefficients is the number of sign changes, or an even amount less than the number of sign changes
  2. TO FIND, JUST COUNT THE NUMBER OF SIGN CHANGES IN THE EQUATION
    1. DON’T FORGET:
      1. P(x) must be in DESCENDING ORDER
      2. Missing powers are OK
    2. Ex: How many positive real roots in $P\left(x\right)=5x^4+3x^3+2x^2-7x+1$
      1. There’s two sign changes between $2x^2-7x$ and $-7x+1$ , so there’s 2 or 0 positive real roots
    3. Ex: How many positive real roots in $P\left(x\right)=x^7-x^6+x^5-x^4+x^3-x^2+x-1$
      1. There’s 7 sign changes, so there’s 7, 5, 3, or 1 positive real roots (+RR)
2. Negative Roots
  1. The number of negative REAL roots of P(x) with REAL coefficients is the number of sign changes, or an even amount less than the number of sign changes, of P(-x)
  2. TO FIND THEM SWITCH SIGNS OF ODD POWERS AND COUNT AGAIN
    1. Ex: How many negative real roots are in $P\left(x\right)=2x^4-3x^5+x^2-1$
      1. First we need to rearrange the sequence so it’s descending, then switch the sign on the $3x^5$ to P(x) $P\left(x\right)=3x^5+2x^4+x^2-1$
      2. Then we can count the sign changes, and there’s only one, so there’s 1 negative real root
3. PRACTICE HOW TO FIND COMPLEX ROOTS TOO

3.7 - Graphing Poly’s and Transformations

Steps for graphing:

Determine the general shape (degree, leading coefficient→END behavior, bumps)
Symmetry (even or odd poly)
Find intercepts (Descartes’ and Rational roots theorem)
1. If they have a multiplicity that is:
  1. Even: the graph bounces
  2. One: the graph crosses to the other side of the x-axis
  3. Odd (other than one): the graph flexes but then crosses over

REVIEW NOTES AND STUFF