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

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

Flashcard 3: 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 4: 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 5: 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 6: 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.

Flashcard 7: 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 8: 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 9: 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 10: 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 11: 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 12: 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 13: 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 14: 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 15: 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 16: 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 17: 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 18: 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 19: 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 20: 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 21: 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 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: 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 24: 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 25: 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 26: 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 27: 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 28: 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 29: 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 30: 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 $x=0$.