4. Higher Degree Polynomials

Question 1

If the degree n is an even number…

Answer 1

both ends of the graph leave the coordinate system at either the top or bottom.

Question 2

If the degree n is an odd number…

Answer 2

the ends of the graph leave the coordinate system at opposite ends.

Question 3

Multiplicity

– the number of times a particular value appears as a root of a polynomial.

Answer 3

The sum of multiplicities of a polynomial is equal to the number of roots of a ploynomial. These are both equal to the degree of the polynomial.

Question 4

Sum of cubes

a³ + b³ =

Answer 4

(a + b)•(a² – ab + b²)

Question 5

Difference of cubes

a³ – b³ =

Answer 5

(a – b)•(a² + ab + b²)

Question 6

Rational Root Theorem

Answer 6

Given a polynomial with integer coefficients, the possible zeroes are found by listing the factors of the constant term over the factors of the leading coefficient, thus forming a list of fractions. This listing gives you a list of potential rational (fractional) roots to test.

Question 7

Factor Theorem

Answer 7

If r is a zero of P(x), then x–r is a factor.

If you synthetic-divide a polynomial by x=r and get a zero remainder, then, not only is x=r a zero of the polynomial, but x–r is also a factor of the polynomial.

Question 8

Sythetic Division

x³–8x–3 / x–3 = ?

Answer 8

3| 1 0 -8 -3

3 9 3

1 3 1 0

= x² + 3x + 1