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, 13x2, x3y2, 5x3+3x2+2x-4• Term– Individual monomials in the polynomial

Order & Identification

• Degree of Polynomial– Largest sum of exponents in single term– Examples:• x4 is degree 4, x3y is degree 4, x4y3z is degree 8

• Written in STANDARD FORM– Largest degree first, then next smaller, etc.

• Coefficient of leading term in standard form is lead coefficient

ClassificationDEGREE NAME EXAMPLE ROOTS

0 Constant 5 0

1 Linear mx + b 1

2 Quadratic ax2+bx+c 2

3 Cubic ax3+bx2+cx+d 3

4 Quartic ax4+bx3+cx2+… 4

5 Quintic ax5+bx4+… 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) = 6x3 + x2 – 5x + 1

– g(x) = x4 – 3

Multiplying Polynomials

• Monomials– Multiply constants– Like variables – add exponents– Unlike variables – combine • Examples: 4*4x , 2x2*3x3 , 3x2*2y2

• Monomial with multi-term polynomial– Distribute• Example: 4x*(x2-3x+2)

Multiplying Polynomials (cont)• Binomial with multi-term– Distribute one term at a time – Then combine like terms• Example: (x-2)(x2-4x+3)

• Multi-term with multi-term– Distribute each term then combine like terms– May help using a box or table to combine• Example: (x2+3x-4) (x2-4x+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

• ____ ____ ____ ____ ____ • 1 4 6 4 1• x x x x x• 3 3 3 3 3

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 2nd coeff• Add 2nd 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: (x3-4x2+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: (4x2 + 3x3 + 10) ÷ (x – 2)

• (15x2 + 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: x3 + 3x2 – 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: (x3 + x2 – 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 Y1 =• 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:– a3 + b3 = (a + b) * (a2 – ab + b2)– a3 – b3 = (a – b) * (a2 + ab + b2)

• Example: x3 + 8

• Example: 2x4 – 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 -2 -1 0 1 2 3

y 22 16 10 4 -2 -8

Year 1950 1960 1970 1980 1990 2000

Population 2853 4011 5065 6720 9708 14,759