Graphing Polynomial Functions Graphs of Polynomial Functions 1

Graphing Polynomial Functions

Graphs of Polynomial Functions • 1. Polynomials have smooth, continuous curves • 2. Continuous means it can be drawn without picking up a pencil • 3. Smooth means it have no sharp points

• 1. Even powers are tangent to the x-axis • 2. Odd powers go across the x-axis at the origin • 3. As the power increases, the curve gets wider at the point of tangency or intersection

Steps for graphing polynomial functions: • • 1. find zeroes 2. Make a number line 3. determine positive and negative values 4. sketch graph

Examples •

Double Roots •

Polynomial Division •

Long Division • Check for placeholders, leave variables • Change signs

Synthetic Division • Find zeros • Check for placeholders • Divide zero into coefficients of polynomial function (multiply, add, repeat)

Remainder Theorem • Let f be a polynomial function. If f(x) is divdied by x-c, then remainder is f(c) • PROOF • When f(x) is divided by x-c, the remainder must be a constant, r, (because the remainder must have a smaller degree than x-c) so by the division algorithm: • f(x)=q(x)(x-c)+r • In order to find f(c), substitute c in for x • f(c)=q(c)(c-c)+r = q(c)0+r = r • So f(c)=r

Factor Theorem • Let f be a polynomial functions. Then x-c is a factor of f(x) if and only if (iff) f(c)=0 • PROOF (must prove both ways) • 1. If x-c is a factor, then when f(x) is divided by x-c the remainder is 0. By the remainder theorem, f(c)=the remainder, so f(c)=0 • 2. If f(c)=0, the remainder theorem, the remainder when f(x) is divided by x-c is 0. If the remainder is 0, then that means x-c goes into f(x), so x-c is a factor of f(x)

Things you know given f(3)=0 • • • 1. When x=3, y=0 2. (3, 0) is an ordered pair on the graph 3. x-3 is a factor of f(x) 4. when f(x) is divided by x-3, the remainder is 0 5. 3 is a root 6. 3 is an x-intercept 7. 3 is a solution if f(x)=0 8. f(x) touches the x-axis at 3 9. f(x) has a zero at 3 10. if the inverse of f(x) exists, then (0, 3) is on it

Things you know if f(-2)=5 • • • 1. When f(x) is divided by x+2, the remainder is 5 2. x+2 is not a factor of f(x) 3. The point (-2, 5) is on f(x) 4. when x= -2, y=5 5. If the inverse of f(x) exists, then the point (5, 2) is on it 6. -2 is not an x-intercept of f(x) 7. -2 is not a solution to f(x)=0 8. -2 is not a root of f(x) 9. -2 is not a zero of f(x)

Rational Zeros Theorem (P/Q) • =

After you identify possible zeros, find actual • Substitute possible zeros into f(x) OR do synthetic division 1 1 -4 -4 16 1 -3 -7 -9 Remainder is not 0, so 1 isn’t a root 2 1 -4 -4 16 2 -4 -16 1 -2 -8 0 Remainder is 0, so 2 is a root. Take coefficients and continue solving for other roots. -1 1 -4 -4 16 -1 5 -1 1 -5 1 15 Remainder is not 0, so -1 isn’t a root

Fundamental Theorem of Algebra • If f(x) is a polynomial function of degree n, where n>0, f(x) has n zeros in the complex number system • Number of answers= degree

Linear Factorization Theorem •

Irrational Conjugate Theorem •

Complex Conjugate Theorem •

Odd Degree Theorem • Any polynomial with real coefficients and with odd degree must have at least one real zero • (x+2 i)(x-5) 5 is real

Factors of a Polynomial •