Zeros of Polynomials to Construct Graphs - Algebra 2
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What is $f(0)$ for $f(x)=-2(x+1)^2(x-3)$?
What is $f(0)$ for $f(x)=-2(x+1)^2(x-3)$?
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$f(0)=6$. Substitute $x=0$: $-2(1)^2(-3)=6$.
$f(0)=6$. Substitute $x=0$: $-2(1)^2(-3)=6$.
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What is $f(0)$ for $f(x)=(x-2)(x+5)$?
What is $f(0)$ for $f(x)=(x-2)(x+5)$?
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$f(0)=-10$. Substitute $x=0$: $(0-2)(0+5)=-10$.
$f(0)=-10$. Substitute $x=0$: $(0-2)(0+5)=-10$.
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What are the zeros of $f(x)=(2x-3)(x+5)$?
What are the zeros of $f(x)=(2x-3)(x+5)$?
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$x=\frac{3}{2}$ and $x=-5$. Set each factor equal to zero and solve.
$x=\frac{3}{2}$ and $x=-5$. Set each factor equal to zero and solve.
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What are the zeros of $f(x)=(x-2)^2(x+7)$?
What are the zeros of $f(x)=(x-2)^2(x+7)$?
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$x=2$ (mult. $2$) and $x=-7$. The squared factor gives multiplicity 2.
$x=2$ (mult. $2$) and $x=-7$. The squared factor gives multiplicity 2.
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What does the multiplicity of a zero tell you about the factorization of $f(x)$?
What does the multiplicity of a zero tell you about the factorization of $f(x)$?
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It is the exponent on the factor $(x-r)$. Higher exponents indicate repeated roots.
It is the exponent on the factor $(x-r)$. Higher exponents indicate repeated roots.
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If $f(x)$ has a zero $r$ with odd multiplicity, how does the graph behave at $x=r$?
If $f(x)$ has a zero $r$ with odd multiplicity, how does the graph behave at $x=r$?
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It crosses the x-axis at $x=r$. Odd multiplicity means the graph changes sides.
It crosses the x-axis at $x=r$. Odd multiplicity means the graph changes sides.
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If $f(x)$ has a zero $r$ with even multiplicity, how does the graph behave at $x=r$?
If $f(x)$ has a zero $r$ with even multiplicity, how does the graph behave at $x=r$?
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It touches and turns at the x-axis at $x=r$. Even multiplicity means the graph stays on same side.
It touches and turns at the x-axis at $x=r$. Even multiplicity means the graph stays on same side.
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For $f(x)=(x+3)^4(x-1)$, which zero has even multiplicity and what is it?
For $f(x)=(x+3)^4(x-1)$, which zero has even multiplicity and what is it?
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$x=-3$ has even multiplicity $4$. The exponent 4 is even.
$x=-3$ has even multiplicity $4$. The exponent 4 is even.
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For $f(x)=(x+3)^4(x-1)$, which zero has odd multiplicity and what is it?
For $f(x)=(x+3)^4(x-1)$, which zero has odd multiplicity and what is it?
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$x=1$ has odd multiplicity $1$. The exponent 1 is odd.
$x=1$ has odd multiplicity $1$. The exponent 1 is odd.
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What is the y-intercept of $y=f(x)$ in terms of $f$?
What is the y-intercept of $y=f(x)$ in terms of $f$?
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The y-intercept is $(0,f(0))$. Substitute $x=0$ into the function.
The y-intercept is $(0,f(0))$. Substitute $x=0$ into the function.
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What is the maximum possible number of real zeros of a degree $n$ polynomial?
What is the maximum possible number of real zeros of a degree $n$ polynomial?
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At most $n$ real zeros. A polynomial can have complex zeros too.
At most $n$ real zeros. A polynomial can have complex zeros too.
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For $f(x)=(x+2)(x^2+4x+4)$, does the graph cross or bounce at $x=-2$?
For $f(x)=(x+2)(x^2+4x+4)$, does the graph cross or bounce at $x=-2$?
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It crosses at $x=-2$ (odd multiplicity $3$). Odd multiplicity means crossing behavior.
It crosses at $x=-2$ (odd multiplicity $3$). Odd multiplicity means crossing behavior.
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What are the zeros of $f(x)=(x+2)(x^2+4x+4)$?
What are the zeros of $f(x)=(x+2)(x^2+4x+4)$?
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$x=-2$ (mult. $3$). Recognize $x^2+4x+4=(x+2)^2$, so total multiplicity is 3.
$x=-2$ (mult. $3$). Recognize $x^2+4x+4=(x+2)^2$, so total multiplicity is 3.
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What are the zeros of $f(x)=(x-1)(x^2-16)$?
What are the zeros of $f(x)=(x-1)(x^2-16)$?
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$x=1$, $x=-4$, and $x=4$. Factor $x^2-16=(x-4)(x+4)$ first.
$x=1$, $x=-4$, and $x=4$. Factor $x^2-16=(x-4)(x+4)$ first.
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For $f(x)=(x-1)^2(x+3)$, does the graph cross or bounce at $x=-3$?
For $f(x)=(x-1)^2(x+3)$, does the graph cross or bounce at $x=-3$?
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It crosses at $x=-3$. Odd multiplicity 1 causes crossing behavior.
It crosses at $x=-3$. Odd multiplicity 1 causes crossing behavior.
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For $f(x)=(x-1)^2(x+3)$, does the graph cross or bounce at $x=1$?
For $f(x)=(x-1)^2(x+3)$, does the graph cross or bounce at $x=1$?
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It bounces (touches and turns) at $x=1$. Even multiplicity 2 causes bouncing behavior.
It bounces (touches and turns) at $x=1$. Even multiplicity 2 causes bouncing behavior.
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What are the zeros of $f(x)=(x^2-9)(x-4)$?
What are the zeros of $f(x)=(x^2-9)(x-4)$?
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$x=-3$, $x=3$, and $x=4$. Factor $x^2-9=(x-3)(x+3)$ first.
$x=-3$, $x=3$, and $x=4$. Factor $x^2-9=(x-3)(x+3)$ first.
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What are the zeros of $f(x)=-(x+2)(x-2)(x-6)$?
What are the zeros of $f(x)=-(x+2)(x-2)(x-6)$?
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$x=-2$, $x=2$, and $x=6$. Set each factor equal to zero.
$x=-2$, $x=2$, and $x=6$. Set each factor equal to zero.
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How does multiplying $f(x)$ by $-1$ affect its zeros?
How does multiplying $f(x)$ by $-1$ affect its zeros?
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It does not change the zeros. Multiplying by constants doesn't change zero locations.
It does not change the zeros. Multiplying by constants doesn't change zero locations.
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How does multiplying $f(x)$ by a nonzero constant $k$ affect its zeros?
How does multiplying $f(x)$ by a nonzero constant $k$ affect its zeros?
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It does not change the zeros. Constants multiply the output, not the input.
It does not change the zeros. Constants multiply the output, not the input.
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What are the zeros of $f(x)=(x^2+5x)(x-2)$?
What are the zeros of $f(x)=(x^2+5x)(x-2)$?
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$x=0$, $x=-5$, and $x=2$. Factor $x^2+5x=x(x+5)$ first.
$x=0$, $x=-5$, and $x=2$. Factor $x^2+5x=x(x+5)$ first.
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What are the zeros of $f(x)=x(x-1)(x+2)$?
What are the zeros of $f(x)=x(x-1)(x+2)$?
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$x=0$, $x=1$, and $x=-2$. Set each factor equal to zero.
$x=0$, $x=1$, and $x=-2$. Set each factor equal to zero.
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Identify the multiplicity of the zero $x=5$ in $f(x)=(x-5)^3(x+1)$.
Identify the multiplicity of the zero $x=5$ in $f(x)=(x-5)^3(x+1)$.
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Multiplicity $3$. The exponent on $(x-5)$ is 3.
Multiplicity $3$. The exponent on $(x-5)$ is 3.
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Identify the multiplicity of the zero $x=-2$ in $f(x)=-7(x+2)^2(x-4)$.
Identify the multiplicity of the zero $x=-2$ in $f(x)=-7(x+2)^2(x-4)$.
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Multiplicity $2$. The exponent on $(x+2)$ is 2.
Multiplicity $2$. The exponent on $(x+2)$ is 2.
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What is a zero of a polynomial function $f(x)$?
What is a zero of a polynomial function $f(x)$?
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A value $r$ such that $f(r)=0$. The zero is where the polynomial equals zero.
A value $r$ such that $f(r)=0$. The zero is where the polynomial equals zero.
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What is the x-intercept on the graph of $y=f(x)$ that corresponds to a zero $r$?
What is the x-intercept on the graph of $y=f(x)$ that corresponds to a zero $r$?
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The point $(r,0)$. Where the graph crosses the x-axis at zero $r$.
The point $(r,0)$. Where the graph crosses the x-axis at zero $r$.
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What does the Factor Theorem state about $x-r$ and a polynomial $f(x)$?
What does the Factor Theorem state about $x-r$ and a polynomial $f(x)$?
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$f(r)=0$ if and only if $(x-r)$ is a factor of $f(x)$. Connects zeros and factors of polynomials.
$f(r)=0$ if and only if $(x-r)$ is a factor of $f(x)$. Connects zeros and factors of polynomials.
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What are the zeros of $f(x)=(x-4)(x+1)$?
What are the zeros of $f(x)=(x-4)(x+1)$?
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$x=4$ and $x=-1$. Set each factor equal to zero and solve.
$x=4$ and $x=-1$. Set each factor equal to zero and solve.
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What is the maximum number of turning points of a degree $n$ polynomial?
What is the maximum number of turning points of a degree $n$ polynomial?
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At most $n-1$ turning points. Turning points occur between consecutive zeros.
At most $n-1$ turning points. Turning points occur between consecutive zeros.
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Identify the zeros of $f(x)=x^2(x-3)(x+1)$ and state the multiplicity of $x=0$.
Identify the zeros of $f(x)=x^2(x-3)(x+1)$ and state the multiplicity of $x=0$.
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Zeros $x=0$ (mult. $2$), $x=3$, $x=-1$. The factor $x^2$ gives multiplicity 2 at $x=0$.
Zeros $x=0$ (mult. $2$), $x=3$, $x=-1$. The factor $x^2$ gives multiplicity 2 at $x=0$.
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