Algebra Flashcards: Zeros Of Polynomials To Construct Graphs

What this deck covers

This deck focuses on Zeros Of Polynomials To Construct Graphs, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 7: Identify the zeros of $f(x)=(2x-1)(x+4)$.

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

Flashcard 8: What is the end behavior if degree is odd and leading coefficient is negative?

Answer: 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.

Flashcard 9: What is the maximum number of real zeros a degree $n$ polynomial can have?

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

Flashcard 10: What is a zero of a polynomial function $f(x)$?

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

Flashcard 11: Identify whether $f(x)=-(x+2)^2(x-3)$ crosses or touches at $x=-2$.

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

Flashcard 12: Identify the zeros of $f(x)=2x^2-8x$ by factoring.

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

Flashcard 13: What is the end behavior if degree is odd and leading coefficient is positive?

Answer: 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.

Flashcard 14: What is the maximum number of turning points a degree $n$ polynomial can have?

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

Flashcard 15: Identify the zeros of $f(x)=x^2-9$ by factoring.

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

Flashcard 16: Which statement is true if $f(x)=(x-2)(x+3)$ and $x=2$ is a zero?

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

Flashcard 17: Identify the zeros of $f(x)=x^3-4x^2$ by factoring.

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

Flashcard 18: Identify the zeros of $f(x)=-(x)(x+4)^3$.

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

Flashcard 19: What is the end behavior if degree is even and leading coefficient is negative?

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

Flashcard 20: Identify whether $f(x)=-(x+2)^2(x-3)$ crosses or touches at $x=3$.

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

Flashcard 21: Identify the zeros of $f(x)=(x^2-1)(x^2-9)$.

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

Flashcard 22: Identify the x-intercepts of $f(x)=(x-1)(x-1)(x+2)$.

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

Flashcard 23: Identify the zero and multiplicity for $f(x)=(x-1)^3(x+2)$ at $x=1$.

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

Flashcard 24: Identify the zeros of $f(x)=(3x+6)(x-2)$.

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

Flashcard 25: What is the y-intercept of $y=f(x)$ written using $f(0)$?

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

Flashcard 26: What is the multiplicity of a zero $r$ if $(x-r)^k$ is a factor of $f(x)$?

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

Flashcard 27: What is the end behavior if degree is even and leading coefficient is positive?

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

Flashcard 28: Identify the zeros of $f(x)=(x^2-1)(x^2-9)$.

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

Flashcard 29: Which end behavior matches $f(x)=-3(x-1)(x+2)(x-5)$?

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

Flashcard 30: Identify whether $f(x)=-(x+2)^2(x-3)$ crosses or touches at $x=3$.

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

Flashcard 31: Identify the zeros of $f(x)=x^2-6x+9$ by factoring.

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

Flashcard 32: What is the rough graph behavior at a zero with multiplicity $4$?

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

Flashcard 33: What is the end behavior if degree is odd and leading coefficient is negative?

Answer: 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.

Flashcard 34: Identify the zeros of $f(x)=x^2+7x+12$ by factoring.

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

Flashcard 35: What is the leading coefficient of $f(x)=-2(x-4)(x+1)^2$?

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

Flashcard 36: Identify the zeros of $f(x)=x^3+x^2-12x$ given $f(x)=x(x+4)(x-3)$.

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

Flashcard 37: Identify the zeros of $f(x)=x^2(x-1)(x+1)$.

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

Flashcard 38: Identify the zeros of $f(x)=x^2-2x-15$ by factoring.

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

Flashcard 39: Identify the zeros of $f(x)=(x-3)(x+5)$.

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

Flashcard 40: What is the end behavior if degree is odd and leading coefficient is positive?

Answer: 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.

Flashcard 41: Identify the zeros of $f(x)=x^2-9$ by factoring.

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

Flashcard 42: Identify the zero and multiplicity for $f(x)=(x+4)^2(x-6)$ at $x=-4$.

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

Flashcard 43: Identify the zeros of $f(x)=x^3-4x^2$ by factoring.

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

Flashcard 44: Identify whether $f(x)=(x+1)^2(x-3)$ crosses or touches at $x=-1$.

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

Flashcard 45: What is the end behavior of $f(x)=-x^4+5x^2$?

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

Flashcard 46: What does it mean if $(x-r)$ is a factor of $f(x)$?

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

Flashcard 47: What is the y-intercept of $f(x)=(x-2)(x+5)$?

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

Flashcard 48: What is the rough graph behavior at a zero with multiplicity $1$?

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

Flashcard 49: Identify the zeros of $f(x)=x(x-7)(x+2)$.

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

Flashcard 50: What is the x-intercept of the graph of $y=f(x)$ in terms of zeros?

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

Flashcard 51: Identify the zeros of $f(x)=(x-3)(x+5)$.

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

Flashcard 52: What is the leading coefficient of $f(x)=-2(x-4)(x+1)^2$?

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

Flashcard 53: What is the multiplicity of the zero $x=0$ for $f(x)=x^2(x-1)(x+1)$?

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

Flashcard 54: Identify the zeros of $f(x)=(2x-1)(x+4)$.

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

Flashcard 55: What is the Factor Theorem stated using $f(r)$ and $(x-r)$?

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

Flashcard 56: What is the degree of $f(x)=(x-1)^2(x+3)$?

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

Flashcard 57: Identify the zeros of $f(x)=x^2+5x$ by factoring.

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

Flashcard 58: What are the x-intercepts of $f(x)=(x-4)(x+1)(x+1)$?

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

Flashcard 59: What is the end behavior of $f(x)=x^5-2x$?

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

Flashcard 60: Identify the zeros of $f(x)=x^4-16$ by factoring completely over reals.

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

Flashcard 61: Identify the zero and multiplicity for $f(x)=(x-1)^3(x+2)$ at $x=1$.

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

Flashcard 62: What is the y-intercept of $f(x)=(x+4)^3$?

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

Flashcard 63: Which end behavior matches $f(x)=-3(x-1)(x+2)(x-5)$?

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

Flashcard 64: What determines end behavior for a polynomial $f(x)$?

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

Flashcard 65: Identify the zeros of $f(x)=x^2-2x-15$ by factoring.

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

Flashcard 66: Identify whether $f(x)=(x+1)^2(x-3)$ crosses or touches at $x=-1$.

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

Flashcard 67: What is the multiplicity of the zero $x=0$ for $f(x)=x^2(x-1)(x+1)$?

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

Flashcard 68: Identify the zeros of $f(x)=x^2+5x$ by factoring.

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

Flashcard 69: What does it mean if $(x-r)$ is a factor of $f(x)$?

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

Flashcard 70: What is the rough graph behavior at a zero with multiplicity $4$?

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

Flashcard 71: Identify the zero and multiplicity for $f(x)=(x+4)^2(x-6)$ at $x=-4$.

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

Flashcard 72: What is a zero of a polynomial function $f(x)$?

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

Flashcard 73: Which statement is true if $f(x)=(x-2)(x+3)$ and $x=2$ is a zero?

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

Flashcard 74: Identify the zeros of $f(x)=(x-5)^2$ and state the multiplicity.

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

Flashcard 75: Identify whether $f(x)=-(x+2)^2(x-3)$ crosses or touches at $x=-2$.

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

Flashcard 76: What is the maximum number of turning points a degree $n$ polynomial can have?

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

Flashcard 77: What is the end behavior of $f(x)=-x^4+5x^2$?

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

Flashcard 78: Identify the zeros of $f(x)=x(x-7)(x+2)$.

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

Flashcard 79: Identify the zeros of $f(x)=x^3+x^2-12x$ given $f(x)=x(x+4)(x-3)$.

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

Flashcard 80: What is the y-intercept of $f(x)=-(x-1)^2(x+3)$?

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

Flashcard 81: What is the degree of $f(x)=(x-1)^2(x+3)$?

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

Flashcard 82: What is the y-intercept of $f(x)=(x+4)^3$?

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

Flashcard 83: What is the x-intercept of the graph of $y=f(x)$ in terms of zeros?

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

Flashcard 84: Identify the x-intercepts of $f(x)=-(x+2)^2(x-3)$.

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

Flashcard 85: Identify the zeros of $f(x)=x^3-9x$ by factoring.

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

Flashcard 86: What is the end behavior of $f(x)=x^5-2x$?

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

Flashcard 87: Identify the zeros of $f(x)=-(x)(x+4)^3$.

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

Flashcard 88: Identify the zeros of $f(x)=x^2(x-1)(x+1)$.

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

Flashcard 89: What is the y-intercept of $f(x)=(x-2)(x+5)$?

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

Flashcard 90: Identify the x-intercepts of $f(x)=(x-1)(x-1)(x+2)$.

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

Flashcard 91: What is the maximum number of real zeros a degree $n$ polynomial can have?

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

Flashcard 92: What is the end behavior if degree is even and leading coefficient is positive?

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

Flashcard 93: What is the rough graph behavior at a zero with multiplicity $1$?

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

Flashcard 94: What are the x-intercepts of $f(x)=(x-4)(x+1)(x+1)$?

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

Flashcard 95: Identify the zeros of $f(x)=2x^2-8x$ by factoring.

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

Flashcard 96: Identify the zeros of $f(x)=x^2+7x+12$ by factoring.

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

Flashcard 97: Which end behavior matches $f(x)=2(x+1)^2(x-4)^2$?

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

Flashcard 98: Which end behavior matches $f(x)=2(x+1)^2(x-4)^2$?

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

Flashcard 99: Identify the zeros of $f(x)=x^3+2x^2-x-2$ given $f(x)=(x+2)(x+1)(x-1)$.

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

Flashcard 100: What is the end behavior if degree is even and leading coefficient is negative?

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