Solving polynomial equations

Today we will use factoring techniques to solve polynomial equations of higher order. The degree of the equation will help determine the number of roots.

Degree of a polynomial

• We determine the degree by arranging the equation in standard form and look at the largest exponent.

Polynomial Function- BackgroundPattern Development

Linear Function: f(x) = ax + b, a 02Quadratic Function: f(x) = ax +bx+c, a 0

3 2Cubic Function: f(x) = ax +bx +cx+d, a 0

4 3 2Quartic Function: f(x) = ax +bx +cx +dx+e, a 0

Degree = 1

Degree = 2

(3)

(4)

The Number of Real Zeros of a Polynomial Function

Polynomial DegreeMinimum # of

Real ZerosMaximum # of

Real Zeros

Linear 1 1 1

Quadratic 2 0 2

Cubic 3 1 3

Quartic 4 0 4

Quintic 5 1 5

Some roots can be repeated

Finding the roots with algebra

• We can try factoring to find roots…• Examples:

1) f (x) =x3 +6x2

2) f(x) =x3 +2x2 + x

Finding the roots with algebra

• We can try factoring to find roots…• Examples:

1) f (x) =x3 +6x2

x2 (x+6)=0x=0,−6

2) f(x) =x3 +2x2 + x

x(x2 +2x+1)→ x(x+1)2 =0x=0,−1

Third degree equation

• Factor by grouping:

x3 −4x−5x2 +20 =0

Third degree equation

• Factor by grouping: x3 −4x−5x2 +20 =0x3 −4x−5x2 +20 =0

(x3 −5x2 )−(4x−20) =0

x2 (x−5)−4(x−5) =0

(x2 −4)(x−5) =0(x−2)(x+2)(x−5) =0x=2,−2,5