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Anatomy of Polynomials

Anatomy of Polynomials. Monomial. A function that can be written as a coefficient times x to an integer power. ƒ (x) =x 5 ƒ ( x)=- 2x 4 ƒ (x)=-7.5x 3 ƒ (x)=x ƒ (x)=-2. Polynomials. A function that can be written as a sum of monomials ƒ (x) =x 5 +x 4 + x 3 +x 2 +x+1

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Anatomy of Polynomials

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  1. Anatomy of Polynomials

  2. Monomial • A function that can be written as a coefficient times x to an integer power. ƒ(x)=x5 ƒ(x)=-2x4 ƒ(x)=-7.5x3 ƒ(x)=x ƒ(x)=-2

  3. Polynomials • A function that can be written as a sum of monomials ƒ(x)=x5+x4+x3+x2+x+1 ƒ(x)=-2x4+3x2-3.1x-7 Any cubic Any quadratic Any line

  4. Counting Bends • An odd degree will always have an even number of bends • An even degree will always have an odd number of bends • The most bends that a polynomial can have is degree-1

  5. End Behavior • Let’s think about the function ƒ(x)=-2x3+3x2+3x-1 • Which way are the ends of the graph pointing? • The leading term is -2x3, so for large x, the graph will behave like -2x3. • When x is a very big number, -2x3 is a very negative number, so as x  ∞ƒ(x)-∞

  6. End Behavior • Let’s think about the function ƒ(x)=-2x3+3x2+3x-1 • Which way are the ends of the graph pointing? • as x  ∞ƒ(x)-∞ • The graph has degree 3, so it has two bends. Working backwards, I go from deceasing at large x, to increasing (first bend), to decreasing (second bend)

  7. End Behavior • Let’s think about the function ƒ(x)=-2x3+3x2+3x-1 • Which way are the ends of the graph pointing? • as x  ∞ƒ(x)-∞ • Before my second bend, the graph is decreasing, so as x-∞, ƒ(x)∞

  8. Solution • Let’s think about the function ƒ(x)=-2x3+3x2+3x-1 • Which way are the ends of the graph pointing? • as x  ∞ƒ(x)-∞ • as x-∞, ƒ(x)∞

  9. Which of the following polynomials exhibit end behavior similar to x6? • 5x4-6x2+2x-7 • 5x3-6x2+2x-7 • 3x2-2x+7821 • Both (a) and (b) • Both (a) and (c)

  10. Which of the following polynomials exhibit end behavior similar to x6? x6 5x4 5x3 3x2 E

  11. Consider the following polynomial below and choose the most correct statement that matches: • The graph of p(x) has 4 real roots, a positive constant term and rises as x approaches both positive and negative infinity. • The graph of p(x) has 4 real roots, a positive constant term and falls as x approaches both positive and negative infinity. • The graph of p(x) has 4 real roots, a negative constant term and rises as x approaches both positive and negative infinity. • The graph of p(x) has 4 real roots, a negative constant term and falls as x approaches both positive and negative infinity. • None of the above statements is 100% accurate.

  12. Positive constant term Rises as x±∞ A Four Real Roots

  13. Solving Polynomials for their roots • This is very hard, involves a lot of guesswork, and sometimes can’t be done at all. • There are methods for making it easier. We will learn some next time. • One tool you CAN use right now is factoring. • Remember to make sure one side of your equation is 0 before you factor.

  14. Suppose the revenue from selling x units of a product is given by R(x)=400x-x3. How many units must be sold to have zero dollars in revenue? a) 0 units b) 20 units c) 50 units d) 400 units e) Both (a) and (b)

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