Zeros of Polynomials to Construct Graphs - Algebra 1

1. What is the y-intercept of $y=f(x)$ written using $f(0)$?

   The y-intercept is $(0,f(0))$. Found by substituting $x=0$ into the function.

2. Identify the x-intercepts of $f(x)=-(x+2)^2(x-3)$.

   $(-2,0)$ and $(3,0)$. X-intercepts occur where each distinct zero is located.

3. What does an odd multiplicity zero do to the graph at $x=r$?

   The graph crosses the x-axis at $x=r$. Odd multiplicities cause the graph to pass through.

4. Identify the zeros of $f(x)=(x-5)^2$ and state the multiplicity.

   $x=5$ with multiplicity $2$. Only one distinct zero from the squared factor.

5. Identify the zeros of $f(x)=x^4-16$ by factoring completely over reals.

   $x=2$ and $x=-2$. Factor as difference of squares twice: $(x^2-4)(x^2+4)=(x-2)(x+2)(x^2+4)$.

6. What is the end behavior if degree is odd and leading coefficient is negative?

   As $x\to-\infty$, $f(x)\to\infty$; as $x\to\infty$, $f(x)\to-\infty$. Odd degree with negative lead: left up, right down.

7. What is the maximum number of real zeros a degree $n$ polynomial can have?

   At most $n$ real zeros. Fundamental Theorem of Algebra applied to real zeros.

8. Identify whether $f(x)=-(x+2)^2(x-3)$ crosses or touches at $x=-2$.

   Touches and turns at $x=-2$. Even multiplicity $2$ means touch and turn behavior.

9. Identify the zeros of $f(x)=x^3-4x^2$ by factoring.

   $x=0$ (mult. $2$) and $x=4$. Factor out $x^2$: $x^2(x-4)=0$.

10. Identify the zeros of $f(x)=-(x)(x+4)^3$.

    $x=0$ and $x=-4$ (mult. $3$ at $-4$). Each distinct zero creates an x-intercept point.

11. What is the end behavior if degree is even and leading coefficient is negative?

    As $x\to\pm\infty$, $f(x)\to-\infty$. Even degree with negative lead: both ends go down.

12. Identify whether $f(x)=-(x+2)^2(x-3)$ crosses or touches at $x=3$.

    Crosses the x-axis at $x=3$. Odd multiplicity $1$ means crossing behavior.

13. Identify the x-intercepts of $f(x)=(x-1)(x-1)(x+2)$.

    $(1,0)$ and $(-2,0)$. X-intercepts occur at distinct zero values.

14. Identify the zeros of $f(x)=(3x+6)(x-2)$.

    $x=-2$ and $x=2$. Factor out the common factor: $3(x+2)(x-2)=0$.

15. What is the multiplicity of a zero $r$ if $(x-r)^k$ is a factor of $f(x)$?

    Multiplicity is $k$. The power of the factor $(x-r)$ in the factorization.

16. Identify the zeros of $f(x)=(x^2-1)(x^2-9)$.

    $x=\pm^1$ and $x=\pm^3$. Set each factor equal to zero and solve.

17. Which end behavior matches $f(x)=-3(x-1)(x+2)(x-5)$?

    As $x\to-\infty$, $f(x)\to\infty$; as $x\to\infty$, $f(x)\to-\infty$. Degree $3$ (odd) with negative leading coefficient.

18. Identify the zeros of $f(x)=x^2-6x+9$ by factoring.

    $x=3$ (multiplicity $2$). Perfect square trinomial: $(x-3)^2=0$.

19. What is the rough graph behavior at a zero with multiplicity $4$?

    The graph touches and turns (does not cross). Even multiplicities create touching/turning behavior.

20. Identify the zeros of $f(x)=x^2+7x+12$ by factoring.

    $x=-3$ and $x=-4$. Factor the quadratic: $(x+3)(x+4)=0$.

21. Identify the zeros of $f(x)=x^3+x^2-12x$ given $f(x)=x(x+4)(x-3)$.

    $x=0$, $x=-4$, and $x=3$. Use the given factorization to identify zeros.

22. Identify the zeros of $f(x)=x^2-2x-15$ by factoring.

    $x=5$ and $x=-3$. Factor the quadratic: $(x-5)(x+3)=0$.

23. Identify the zeros of $f(x)=(x-3)(x+5)$.

    $x=3$ and $x=-5$. Set each factor equal to zero and solve.

24. What is the end behavior if degree is odd and leading coefficient is positive?

    As $x\to-\infty$, $f(x)\to-\infty$; as $x\to\infty$, $f(x)\to\infty$. Odd degree with positive lead: left down, right up.

25. Identify the zeros of $f(x)=x^2-9$ by factoring.

    $x=3$ and $x=-3$. Factor as difference of squares: $(x-3)(x+3)$.

26. Identify the zero and multiplicity for $f(x)=(x+4)^2(x-6)$ at $x=-4$.

    Zero $x=-4$ with multiplicity $2$. The factor $(x+4)^2$ gives zero $x=-4$ with multiplicity $2$.

27. Identify whether $f(x)=(x+1)^2(x-3)$ crosses or touches at $x=-1$.

    Touches and turns at $x=-1$. Even multiplicity $2$ means the graph touches and turns.

28. What is the end behavior of $f(x)=-x^4+5x^2$?

    As $x\to\pm\infty$, $f(x)\to-\infty$. Even degree $4$ with negative leading coefficient.

29. What is the rough graph behavior at a zero with multiplicity $1$?

    The graph crosses the x-axis. Odd multiplicities create crossing behavior.

30. Identify the zeros of $f(x)=x(x-7)(x+2)$.

    $x=0$, $x=7$, and $x=-2$. Set each factor equal to zero and solve.

31. What is the x-intercept of the graph of $y=f(x)$ in terms of zeros?

    Any point $(r,0)$ where $r$ is a zero of $f(x)$. Zeros create x-intercepts where the graph crosses the x-axis.

32. What is the leading coefficient of $f(x)=-2(x-4)(x+1)^2$?

    Leading coefficient $-2$. Coefficient of the highest degree term when expanded.

33. Identify the zeros of $f(x)=(2x-1)(x+4)$.

    $x=\frac{1}{2}$ and $x=-4$. Set each factor equal to zero: $2x-1=0$ and $x+4=0$.

34. What is the end behavior of $f(x)=x^5-2x$?

    As $x\to-\infty$, $f(x)\to-\infty$; as $x\to\infty$, $f(x)\to\infty$. Odd degree $5$ with positive leading coefficient.

35. Identify the zero and multiplicity for $f(x)=(x-1)^3(x+2)$ at $x=1$.

    Zero $x=1$ with multiplicity $3$. The factor $(x-1)^3$ gives zero $x=1$ with multiplicity $3$.

36. What is the y-intercept of $f(x)=(x+4)^3$?

    $(0,64)$. Substitute $x=0$: $f(0)=(0+4)^3=64$.

37. What is the multiplicity of the zero $x=0$ for $f(x)=x^2(x-1)(x+1)$?

    Multiplicity $2$. The factor $x^2$ gives multiplicity $2$ at zero.

38. Identify the zeros of $f(x)=x^2+5x$ by factoring.

    $x=0$ and $x=-5$. Factor out common $x$: $x(x+5)=0$.

39. What does it mean if $(x-r)$ is a factor of $f(x)$?

    $r$ is a zero of $f(x)$. Factor Theorem: $(x-r)$ is a factor iff $r$ is a zero.

40. What is a zero of a polynomial function $f(x)$?

    A value $r$ such that $f(r)=0$. When the function equals zero at that input value.

41. Which statement is true if $f(x)=(x-2)(x+3)$ and $x=2$ is a zero?

    The graph includes the x-intercept $(2,0)$. Zeros correspond to x-intercepts on the graph.

42. What is the maximum number of turning points a degree $n$ polynomial can have?

    At most $n-1$ turning points. Local maxima and minima occur between zeros.

43. What is the y-intercept of $f(x)=-(x-1)^2(x+3)$?

    $(0,-3)$. Substitute $x=0$: $f(0)=-(0-1)^2(0+3)=-3$.

44. What is the degree of $f(x)=(x-1)^2(x+3)$?

    Degree $3$. Count the highest power when expanded.

45. Identify the zeros of $f(x)=x^2(x-1)(x+1)$.

    $x=0$ (mult. $2$), $x=1$, and $x=-1$. Set each factor equal to zero and solve.

46. What is the y-intercept of $f(x)=(x-2)(x+5)$?

    $(0,-10)$. Substitute $x=0$: $f(0)=(0-2)(0+5)=-10$.

47. What is the end behavior if degree is even and leading coefficient is positive?

    As $x\to\pm\infty$, $f(x)\to\infty$. Even degree with positive lead: both ends go up.

48. What are the x-intercepts of $f(x)=(x-4)(x+1)(x+1)$?

    $(4,0)$ and $(-1,0)$. X-intercepts occur where each distinct zero equals zero.

49. Identify the zeros of $f(x)=2x^2-8x$ by factoring.

    $x=0$ and $x=4$. Factor out $2x$: $2x(x-4)=0$.

50. Which end behavior matches $f(x)=2(x+1)^2(x-4)^2$?

    As $x\to\pm\infty$, $f(x)\to\infty$. Degree $4$ (even) with positive leading coefficient.

51. What is the Factor Theorem stated using $f(r)$ and $(x-r)$?

    $(x-r)$ is a factor of $f(x)$ iff $f(r)=0$. The fundamental connection between factors and zeros.

52. Identify the zeros of $f(x)=x^3-9x$ by factoring.

    $x=0$, $x=3$, and $x=-3$. Factor out $x$: $x(x^2-9)=x(x-3)(x+3)=0$.

53. Identify whether $f(x)=(x-2)^5(x+7)$ crosses or touches at $x=2$.

    Crosses the x-axis at $x=2$. Odd multiplicity $5$ means the graph crosses the axis.

54. What does an even multiplicity zero do to the graph at $x=r$?

    The graph touches and turns at $x=r$. Even multiplicities create a bounce off the x-axis.

55. Identify the zeros of $f(x)=x^3+2x^2-x-2$ given $f(x)=(x+2)(x+1)(x-1)$.

    $x=-2$, $x=-1$, and $x=1$. Use the given factorization to find zeros.

56. What determines end behavior for a polynomial $f(x)$?

    The leading term (degree and leading coefficient). The highest degree term controls behavior as $x\to\pm\infty$.

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