Algebra 2 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 2.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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