2 1 Polynomial Function fguilbert Objective to identify
2. 1 Polynomial Function fguilbert
Objective: to identify a polynomial function, to evaluate using syntheti synthet substitution and to determ its zeros. fguilbert
Form of a polynomial anx + an-1 x +. . . a 1 + ao 7 6 2 1 5 x - 2 x +. . . +5 x - 4 x + n terms fguilbert n-1 1 x
Form of a polynomial anx + an-1 x +. . . a 1 + ao 7 6 2 1 5 x - 2 x +. . . +5 x - 4 x + n n-1 coefficients fguilbert 1 x
The term with the larges exponent is usually first. 3 5 x +x-8 5 The leading coefficient is _ fguilbert 2 3 x
Degree Name 0 constant 1 linear 2 quadratic 3 cubic 4 quartic 15 quintic fguilbert E 2 2 x +1 + 2 x 3 2 x +x 4 7 x +5 5 3 x -x
If P(x) = 0, x is a ROOT and a ZERO of the functio Slight difference between zero and a root. Zero refers to a function. fguilbert
2 is a zero of the polynom 2 P(x) = x - 4 2 is a root of the 2 0=x -4 fguilbert
ex 1. P(x) = -7 3 x -5 Find P(-2) substitute -2 for x fguilbert 2 x +10 +1
ex 1. P(x) = -7 3 x -5 2 x +10 +1 Remember use ( ) when substituting fguilbert
ex 1. P(x) = -7 -5 +1 2 P(-2)=-7 (-2 )3 -5( -2 ) +10 = -7 ( -8) -5( 4) + 10 = 56 - 20 + 10 = 46 fguilbert 3 x 2 x
ex 2. 3 x P(x) = -7 -5 Find P(2 n) Substitute 2 n for x USE ( ) when substituting fguilbert 2 x +1
ex 1. 3 x P(x) = -7 -5 3 2 n P(2 n)=-7 (2 n ) -5( 3 (8 n ) 2 x +10 2 ) +1 2 4 n ) -5( +1 = -7 3 2 = -56 n - 20 n + 10 ou MUST know Exponent Rules Rule fguilbert
P(x) = 3 -7 x 3 (-7 x -5 2 x +10 2 x IS -5 +10) Divisible by (x+2) You need to know the If the remainder is 0, remainder. then x+2 ix a factor. fguilbert
P(x) = -7 3 x -5 2 x +10 divide by (x+2)no X term -2 l - 7 - 5 0 10 Notice that to check (x +2) you divide by - 2. fguilbert
P(x) = -7 -2 l fguilbert -7 3 x -5 2 x -5 14 -18 -7 9 -18 +10 0 10 36 46
The remainder is the same with synthetic substitution. P (-2) = 46. fguilbert
P(x) = -7 3 x -5 2 x +10 P( -2 ) gives remainder 46 If remainder is NOT 0 the X + 2 is NOT a factor. fguilbert
So how do I check to se 3 2 if P(x) = -7 x -5 x +10 is divisible by x +2 ? ? ? Either Synthetic division Synthetic substitution If r = 0, it’s divisible. fguilbert
3 x 2 is a zero of f(x)= 2 FIND K 3 0=2 x + kx 3 0 = 2 2( ) +2 k( 0 = 2 ( 8) + 2 k 0 = 10 + 2 k -5 = k fguilbert +k -6 )-
thought for` the day Winners never quit. fguilbert
- Slides: 21