# 3.3e: Exercises - Polynomial End Behaviour

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### A: Concepts

Exercise $$\PageIndex{A}$$

1) Explain the difference between the coefficient of a power function and its degree.

2) If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?

3) In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.

4) What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As $$x \rightarrow-\infty, f(x) \rightarrow-\infty$$ and as $$x \rightarrow \infty, f(x) \rightarrow-\infty$$.

1. The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.

3. As $$x$$ decreases without bound, so does $$f(x)$$. As $$x$$ increases without bound, so does $$f(x)$$.

### B: Identify Power Functions and Polynomials

Exercise $$\PageIndex{B}$$

$$\bigstar$$ Identify the function as a power function, a polynomial function, or neither. If neither, explain.

 5) $$f(x)=(x^2)^3$$ 6) $$f(x)=x^5$$ 7) $$f(x)=x−x^4$$ 8) $$f(x)=\dfrac{x^2}{x^2−1}$$ 9) $$f(x)=3^{x+1}$$ 10) $$f(x)=2x(x+2)(x−1)^2$$

5. Power function, $$\quad$$ 7. Polynomial, $$\quad$$ 9. Neither - this is an exponential function because the variable is in the exponent

### C: Degree and Leading Coefficient of a Polynomial

Exercise $$\PageIndex{C}$$

$$\bigstar$$ (a) Find the degree and leading coefficient for the given polynomial. (b) State the end behaviour.

 11) $$7−2x^2$$ 12) $$−3x^4$$ 13) $$x(4−x^2)(2x+1)$$ 14) $$−2x^2− 3x^5+ x−6$$ 15) $$x^2(2x−3)^2$$ 16) $$-3(2x+5)^2(1-x)^3$$
 11. Degree $$2$$, Coefficient = $$-2$$ $$\swarrow \dots \searrow$$ 13. Degree $$4$$, Coefficient = $$-2$$ $$\swarrow \dots \searrow$$ 15. Degree $$4$$, Coefficient = $$4$$  or  $$\nwarrow \dots \nearrow$$

### D: Polynomial End Behaviour

Exercise $$\PageIndex{D}$$

$$\bigstar$$ Determine the end behavior of the functions.

 17) $$f(x)=x^4$$ 18) $$f(x)=x^3$$ 19) $$f(x)=−x^4$$ 20) $$f(x)=−x^9$$ 21) $$f(x)=−2x^4− 3x^2+ x−1$$ 22) $$f(x)=3x^2+ x−2$$ 23) $$f(x)=x^2(2x^3−x+1)$$ 24(a) $$f(x)=(2−x)^7$$ 24(b) $$f(x)=3x(2x-3)^4$$ 25) $$f(x)=x^4−5x^2$$ 26) $$f(x)=−x^3$$ 27) $$f(x)=(x−1)(x−2)(3−x)$$ 28) $$f(x)=x^2(1−x)^2$$ 29) $$f(x)=\dfrac{x^5}{10}−x^4$$
 17. As $$x→\pm ∞$$, $$f(x)→∞$$, $$\quad$$or  $$\nwarrow \dots \nearrow$$ 19. As $$x→\pm ∞$$, $$f(x)→−∞$$, $$\quad$$or  $$\swarrow \dots \searrow$$ 21. As $$x→\pm∞$$, $$f(x)→−∞$$, $$\quad$$or  $$\swarrow \dots \searrow$$ 23. As $$x→∞$$, $$f(x)→∞$$, as $$x→−∞$$, $$f(x)→−∞$$ $$\quad \;$$ or  $$\swarrow \dots \nearrow$$ 25. As $$x→\pm ∞$$, $$f(x)→∞$$, or  $$\nwarrow \dots \nearrow$$ 27. as $$x→−∞,$$ $$f(x)→∞$$, as $$x→∞,$$ $$f(x)→−∞$$ $$\quad \;$$ or  $$\nwarrow \dots \searrow$$ 29.  As $$x→∞$$, $$f(x)→∞$$, as $$x→−∞$$, $$f(x)→−∞$$ $$\quad \;$$ or  $$\swarrow \dots \nearrow$$

### E: Polynomial Intercepts

Exercise $$\PageIndex{E}$$

$$\bigstar$$ Find the intercepts of the functions.

 30) $$C(t)=2(t−4)(t+1)(t−6)$$ 31) $$f(t)=2(t−1)(t+2)(t−3)$$ 32) $$g(n)=−2(3n−1)(2n+1)$$ 33) $$f(x)=x^4−16$$ 34) $$f(x)=x^3+27$$ 35) $$f(x)=x(x^2−2x−8)$$ 36) $$f(x)=(x+3)(4x^2−1)$$ 37) $$C(t)=3(t+2)(t−3)(t+5)$$ 38) $$C(t)=4t(t−2)^2(t+1)$$ 39) $$C(t)=2t(t−3)(t+1)^2$$ 40) $$C(t)=2t^4−8t^3+6t^2$$ 41) $$C(t)=4t^4+12t^3−40t^2$$ 42) $$f(x)=x^4−x^2$$ 43) $$f(x)=x^3+x^2−20x$$ 44) $$f(x)=x^3+6x^2−7x$$ 45) $$f(x)=x^3+x^2−4x−4$$ 46) $$f(x)=x^3+2x^2−9x−18$$ 47) $$f(x)=2x^3−x^2−8x+4$$ 48) $$f(x)=x^6−7x^3−8$$ 49) $$f(x)=2x^4+6x^2−8$$ 50) $$f(x)=x^3−3x^2−x+3$$ 51) $$f(x)=x^6−2x^4−3x^2$$ 52) $$f(x)=x^6−3x^4−4x^2$$ 53) $$f(x)=x^5−5x^3+4x$$

$$\bigstar$$ Determine the intercepts and the end behavior.

 55) $$f(x)=x^3(x−2)$$ 56) $$f(x)=x(x−3)(x+3)$$ 57) $$f(x)=x(14−2x)(10−2x)$$ 58) $$f(x)=x(14−2x)(10−2x)^2$$ 59) $$f(x)=x^3−16x$$ 60) $$f(x)=x^3−27$$ 61) $$f(x)=x^4−81$$ 62) $$f(x)=−x^3+x^2+2x$$ 63) $$f(x)=x^3−2x^2−15x$$ 64) $$f(x)=x^3−0.01x$$
 31.$$(0,12)$$; $$(1,0),\;(–2,0),\; (3,0)$$ 33. $$(0,−16)$$; $$(2,0),\; (−2,0)$$. 35. $$(0,0), \; (4,0), \; (−2, 0)$$. 37. $$(0, 90)$$; $$(−2,0),\;(3,0),\;(−5,0)$$ 39. $$(0,0)$$, $$(3,0),(−1,0)$$ 41. $$(0,0), (−5,0),\; (2,0)$$ 43. $$(0,0),\; (−5,0),\; (4,0)$$ 45. $$(0, -4)$$; $$(2,0),\; (−2,0),\; (−1,0)$$ 47. $$(0, 4)$$;  $$(−2,0),\;(2,0),\left(\tfrac{1}{2},0\right)$$ 49. $$(0, -8)$$; $$(1,0),\;(−1,0)$$ 51. $$(0,0),\;(\sqrt{3},0)),\;(−\sqrt{3},0)$$ 53. $$(0,0),\,(1,0),(−1,0),(2,0),(−2,0)$$

55. $$(0, 0)$$; $$(0, 0), (2, 0)$$.  As $$x→\pm ∞,$$ $$f(x)→∞$$

57. $$(0,0)$$;  $$(0, 0), (5, 0), (7, 0)$$. As $$x→−∞$$, $$f(x)→−∞$$, as $$x→∞,$$ $$f(x)→∞$$

59. $$(0, 0)$$; $$(−4, 0), (0, 0), (4, 0)$$. As $$x→−∞$$, $$f(x)→−∞$$, as $$x→∞,$$ $$f(x)→∞$$

61. $$(0, -81)$$; $$(3, 0), (−3, 0)$$. As $$x→\pm ∞,$$ $$f(x)→∞$$

63.$$(0, 0)$$; $$(−3, 0), (0, 0), (5, 0).$$ As $$x→−∞$$, $$f(x)→−∞$$, as $$x→∞,$$ $$f(x)→∞$$

$$\star$$

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