Polynomials

Overview Definition – 1 or more terms, each term consisting of a constant multiplier and one or more variables raised to nonnegative integral powers – Examples: 10, 13x 2, x 3 y 2, 5x 3 +3x 2 +2x-4 Term – Individual monomials in the polynomial

Order & Identification Degree of Polynomial – Largest sum of exponents in single term – Examples: x 4 is degree 4, x 3 y is degree 4, x 4 y 3 z is degree 8 Written in STANDARD FORM – Largest degree first, then next smaller, etc. Coefficient of leading term in standard form is lead coefficient

Classification DEGREENAMEEXAMPLEROOTS 0Constant50 1Linearmx + b1 2Quadraticax 2 +bx+c2 3Cubicax 3 +bx 2 +cx+d3 4Quarticax 4 +bx 3 +cx 2 +…4 5Quinticax 5 +bx 4 +…5

Adding & Subtracting Place polynomials in standard form Add or subtract LIKE TERMS – EXACT SAME variables to same powers When adding or subtracting – VARIABLE EXPONENTS STAY THE SAME – Coefficients are changed

Graphing Place polynomials in standard form Insert function (polynomial) into Y= Shows behavior of polynomial (what to expect) – End States – Domain & Range (Minimum or Maximum) – Real Roots (or Zeros) Examples: – f(x) = 6x 3 + x 2 – 5x + 1 – g(x) = x 4 – 3

Multiplying Polynomials Monomials – Multiply constants – Like variables – add exponents – Unlike variables – combine Examples: 4*4x, 2x 2 *3x 3, 3x 2 *2y 2 Monomial with multi-term polynomial – Distribute Example: 4x*(x 2 -3x+2)

Multiplying Polynomials (cont) Binomial with multi-term – Distribute one term at a time – Then combine like terms Example: (x-2)(x 2 -4x+3) Multi-term with multi-term – Distribute each term then combine like terms – May help using a box or table to combine Example: (x 2 +3x-4) (x 2 -4x+3)

Multiplying Polynomials (cont) Binomial raised to a power – Expand out the binomials – FOIL 2 binomials - repeat if required – Multiply result times binomial or use a box Example: (x + 2) 4

Simplification - Multiplying Binomials Pascal’s Triangle – used for (a + b) n – Quick expansion of binomials raised to a power – There will always be n + 1 terms – Lead exponent will be n

Example of Pascal’s Triangle Expand (x + 3) 4 ____ ____ ____ ____ ____ x x x x x

Dividing Polynomials Synthetic Division - Shorthand method of dividing polynomial by binomial using the coefficients Find a factor – then the root Write coefficients & root in synthetic division format Bring down first coefficient Multiply root * coefficient : product under 2 nd coeff Add 2 nd coefficient and product – bring down sum Continue across all coefficients – Insert zero where exponents leave a gap Number under the constant term is remainder

Remainder Theorem If polynomial, P(x), is divided by factor (x-a), then the remainder after division in the value of the polynomial for the value of that root – r = P(a) – Example: (x 3 -4x 2 +5x+1)÷(x-3) If remainder = zero: factor is a root (solution)

Long Division Lead Coefficient not 1 Lead variable exponent not 1 Done same way as regular long division Examples: (4x 2 + 3x ) ÷ (x – 2) (15x 2 + 8x – 12) ÷ (3x + 1)

Factoring by Grouping For a polynomial with 4 terms – Group the first two terms and last two terms – Pull out common factors from each new group – Look for common factor/remainder – Continue factoring if able (Difference of Squares) Example: x 3 + 3x 2 – 4x – 12

Factoring If a divisor (given factor) has a remainder of 0 – The factor is a root of the polynomial Using Synthetic Division – Divide through and reduce the initial polynomial Factor resulting quadratic Example: (x 3 + x 2 – 10x + 8) ÷ (x – 2)

Factors to Roots SAME AS IN QUADRATICS!!!! Find the factors Set the factor equal to 0 (i.e. Factor x-1 = 0) Then isolate the x

Using a Calculator to Find Roots Degree determines number of roots Enter the polynomial in Y 1 = Look at GRAPH to see if the polynomial crosses the x axis – this is a real root – May have to change window or zoom – If touches – double root at that point – If doesn’t cross but bends – imaginary roots Look at TABLE to determine if roots are integers

More Roots Once you find one root: – Use synthetic division to find new equation – Factor new equation (if able) – Look at calculator to find more roots – Use synthetic division again to find more factors Example: Gronk & the Glove Pizzazz

Sum or Difference of Cubes Special rule for sum or difference of 2 cubes: – a 3 + b 3 = (a + b) * (a 2 – ab + b 2 ) – a 3 – b 3 = (a – b) * (a 2 + ab + b 2 ) Example: x Example: 2x 4 – 54x

Root Theorems Rational Root Theorem – If a polynomial has integer coefficients: Every Rational Root can be written (found) by p/q – P is the factors of the constant – Q is the factors of the lead coefficient Irrational Root Theorem – If a polynomial has a + b√c as a root: Then a − b√c is also a root – Same is true for imaginary roots ( both ± ai )

End State Behavior Lead Coefficient - where P(x) is going (x  +∞) Degree of Lead Term – where P(x) came from (x  -∞) – First check lead coefficient - Final Direction + a : Final End Up – a : Final End Down – Then check degree Even: initial P(x) matches final: Up – Up or Down – Down Odd: initial P(x) opposite final: Up – Down or Down – Up Example:

Transformations Transformations are the same as quadratics – f(x - h) – f(x) + k – a f(x) – f(a x) – – f(x) – f(–x)

Using Data to Determine Degree Finite differences - differences between y values – Subtract previous from latter starting on the right – Look if differences are constant (or almost) – If not, try again with the new numbers – This determines which type of model (polynomial) best represents data First time through – first order – linear Second time through – second order – quadratic Third time through – third order – cubic (Degree – 3) Etc.,

Example of Data Modeling Data 1: Data 2: x y Year Population ,759