Zeros of Polynomials to Construct Graphs - Algebra 2

1. For $f(x)=x^2(x-3)(x+1)$, does the graph cross or bounce at $x=0$?

   It bounces at $x=0$ (even multiplicity $2$). Even multiplicity causes bouncing behavior.

2. How do you determine end behavior from a polynomial written in standard form $a_nx^n+\cdots$?

   Use the degree $n$ and leading coefficient $a_n$. These determine the polynomial's long-term behavior.

3. For $f(x)=(x-1)^2(x+3)$, does the graph cross or bounce at $x=-3$?

   It crosses at $x=-3$. Odd multiplicity 1 causes crossing behavior.

4. What does the multiplicity of a zero tell you about the factorization of $f(x)$?

   It is the exponent on the factor $(x-r)$. Higher exponents indicate repeated roots.

5. For $f(x)=-3(x-2)^2(x+1)$, what is the end behavior as $x\to-\infty$?

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

6. If $f(x)$ has even degree and leading coefficient $a_n>0$, what is the end behavior?

   As $x\to\pm\infty$, $f(x)\to\infty$. Both ends go up for positive even-degree polynomials.

7. Identify the multiplicity of the zero $x=5$ in $f(x)=(x-5)^3(x+1)$.

   Multiplicity $3$. The exponent on $(x-5)$ is 3.

8. For $f(x)=2(x-1)(x+4)^2$, what is the end behavior as $x\to\infty$?

   As $x\to\infty$, $f(x)\to\infty$. Degree 3 with positive leading coefficient.

9. What is $f(0)$ for $f(x)=-2(x+1)^2(x-3)$?

   $f(0)=6$. Substitute $x=0$: $-2(1)^2(-3)=6$.

10. What is the factored form of $x^2+6x+9$ useful for finding zeros?

    $(x+3)^2$. Perfect square trinomial formula.

11. What are the zeros of $f(x)=(x^2+5x)(x-2)$?

    $x=0$, $x=-5$, and $x=2$. Factor $x^2+5x=x(x+5)$ first.

12. Identify the multiplicity of the zero $x=-2$ in $f(x)=-7(x+2)^2(x-4)$.

    Multiplicity $2$. The exponent on $(x+2)$ is 2.

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

    A value $r$ such that $f(r)=0$. The zero is where the polynomial equals zero.

14. For $f(x)=(x+4)^2(x-2)^2$, does the graph cross the x-axis at $x=2$?

    No; it touches and turns at $x=2$. Even multiplicity 2 causes bouncing.

15. If $f(x)$ has odd degree and leading coefficient $a_n<0$, what is the end behavior?

    As $x\to-\infty$, $f(x)\to\infty$; $x\to\infty$, $f(x)\to-\infty$. Negative odd-degree polynomials fall from left to right.

16. What is the x-intercept on the graph of $y=f(x)$ that corresponds to a zero $r$?

    The point $(r,0)$. Where the graph crosses the x-axis at zero $r$.

17. What are the zeros of $f(x)=(x^2-4x+4)(x+5)$?

    $x=2$ (mult. $2$) and $x=-5$. Recognize $x^2-4x+4=(x-2)^2$.

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

    $x=4$ and $x=-1$. Set each factor equal to zero and solve.

19. How does multiplying $f(x)$ by a nonzero constant $k$ affect its zeros?

    It does not change the zeros. Constants multiply the output, not the input.

20. If $f(x)$ has a zero $r$ with even multiplicity, how does the graph behave at $x=r$?

    It touches and turns at the x-axis at $x=r$. Even multiplicity means the graph stays on same side.

21. What is the maximum number of turning points of a degree $n$ polynomial?

    At most $n-1$ turning points. Turning points occur between consecutive zeros.

22. What are the zeros of $f(x)=x(x-1)(x+2)$?

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

23. For $f(x)=(x+4)^2(x-2)^2$, what is the end behavior as $x\to\pm\infty$?

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

24. For $f(x)=(x+3)^4(x-1)$, which zero has odd multiplicity and what is it?

    $x=1$ has odd multiplicity $1$. The exponent 1 is odd.

25. What are the zeros of $f(x)=(x-3)(x^2+2x+1)$?

    $x=3$ and $x=-1$ (mult. $2$). Recognize $x^2+2x+1=(x+1)^2$.

26. What are the zeros of $f(x)=(x-2)^2(x+7)$?

    $x=2$ (mult. $2$) and $x=-7$. The squared factor gives multiplicity 2.

27. What are the zeros of $f(x)=(x-1)(x^2-16)$?

    $x=1$, $x=-4$, and $x=4$. Factor $x^2-16=(x-4)(x+4)$ first.

28. What are the zeros of $f(x)=(x^2-9)(x-4)$?

    $x=-3$, $x=3$, and $x=4$. Factor $x^2-9=(x-3)(x+3)$ first.

29. Identify the zeros of $f(x)=x^2(x-3)(x+1)$ and state the multiplicity of $x=0$.

    Zeros $x=0$ (mult. $2$), $x=3$, $x=-1$. The factor $x^2$ gives multiplicity 2 at $x=0$.

30. For $f(x)=(x+3)^4(x-1)$, which zero has even multiplicity and what is it?

    $x=-3$ has even multiplicity $4$. The exponent 4 is even.

31. If $f(x)$ has even degree and leading coefficient $a_n<0$, what is the end behavior?

    As $x\to\pm\infty$, $f(x)\to-\infty$. Both ends go down for negative even-degree polynomials.

32. What are the zeros of $f(x)=(2x-3)(x+5)$?

    $x=\frac{3}{2}$ and $x=-5$. Set each factor equal to zero and solve.

33. What is the factored form of $x^2-9$ useful for finding zeros?

    $(x-3)(x+3)$. Difference of squares formula.

34. What are the zeros of $f(x)=(3x+6)(x^2-1)$?

    $x=-2$, $x=-1$, and $x=1$. Factor $3x+6=3(x+2)$ and $x^2-1=(x-1)(x+1)$.

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

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

36. What does the Factor Theorem state about $x-r$ and a polynomial $f(x)$?

    $f(r)=0$ if and only if $(x-r)$ is a factor of $f(x)$. Connects zeros and factors of polynomials.

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

    Leading coefficient $2$. The coefficient 2 multiplies the entire expression.

38. Identify the zeros of $f(x)=(x+4)^2(x-2)^2$.

    $x=-4$ (mult. $2$) and $x=2$ (mult. $2$). Both zeros have even multiplicity 2.

39. What is the maximum possible number of real zeros of a degree $n$ polynomial?

    At most $n$ real zeros. A polynomial can have complex zeros too.

40. For $f(x)=(x-1)^2(x+3)$, does the graph cross or bounce at $x=1$?

    It bounces (touches and turns) at $x=1$. Even multiplicity 2 causes bouncing behavior.

41. How does multiplying $f(x)$ by $-1$ affect its zeros?

    It does not change the zeros. Multiplying by constants doesn't change zero locations.

42. If $f(x)$ has odd degree and leading coefficient $a_n>0$, what is the end behavior?

    As $x\to-\infty$, $f(x)\to-\infty$; $x\to\infty$, $f(x)\to\infty$. Positive odd-degree polynomials rise from left to right.

43. What is $f(0)$ for $f(x)=(x-2)(x+5)$?

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

44. For $f(x)=(x+2)(x^2+4x+4)$, does the graph cross or bounce at $x=-2$?

    It crosses at $x=-2$ (odd multiplicity $3$). Odd multiplicity means crossing behavior.

45. What is the factored form of $x^2-4x+4$ useful for finding zeros?

    $(x-2)^2$. Perfect square trinomial formula.

46. What is the degree and leading coefficient of $f(x)=-3(x-2)^2(x+1)$?

    Degree $3$, leading coefficient $-3$. Multiply exponents to get degree 3, coefficient is $-3$.

47. For $f(x)=-3(x-2)^2(x+1)$, what is the end behavior as $x\to\infty$?

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

48. What are the zeros of $f(x)=(x+2)(x^2+4x+4)$?

    $x=-2$ (mult. $3$). Recognize $x^2+4x+4=(x+2)^2$, so total multiplicity is 3.

49. What are the zeros of $f(x)=-(x+2)(x-2)(x-6)$?

    $x=-2$, $x=2$, and $x=6$. Set each factor equal to zero.

50. If $f(x)$ has a zero $r$ with odd multiplicity, how does the graph behave at $x=r$?

    It crosses the x-axis at $x=r$. Odd multiplicity means the graph changes sides.

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