Zeros of Polynomials to Construct Graphs - Algebra
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What determines end behavior for a polynomial $f(x)$?
What determines end behavior for a polynomial $f(x)$?
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The leading term (degree and leading coefficient). The highest degree term controls behavior as $x\to\pm\infty$.
The leading term (degree and leading coefficient). The highest degree term controls behavior as $x\to\pm\infty$.
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What is the y-intercept of $y=f(x)$ written using $f(0)$?
What is the y-intercept of $y=f(x)$ written using $f(0)$?
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The y-intercept is $(0,f(0))$. Found by substituting $x=0$ into the function.
The y-intercept is $(0,f(0))$. Found by substituting $x=0$ into the function.
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What does an even multiplicity zero do to the graph at $x=r$?
What does an even multiplicity zero do to the graph at $x=r$?
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The graph touches and turns at $x=r$. Even multiplicities create a bounce off the x-axis.
The graph touches and turns at $x=r$. Even multiplicities create a bounce off the x-axis.
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Identify the zeros of $f(x)=x^2+5x$ by factoring.
Identify the zeros of $f(x)=x^2+5x$ by factoring.
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$x=0$ and $x=-5$. Factor out common $x$: $x(x+5)=0$.
$x=0$ and $x=-5$. Factor out common $x$: $x(x+5)=0$.
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Identify the zeros of $f(x)=x^2-6x+9$ by factoring.
Identify the zeros of $f(x)=x^2-6x+9$ by factoring.
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$x=3$ (multiplicity $2$). Perfect square trinomial: $(x-3)^2=0$.
$x=3$ (multiplicity $2$). Perfect square trinomial: $(x-3)^2=0$.
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What is the multiplicity of a zero $r$ if $(x-r)^k$ is a factor of $f(x)$?
What is the multiplicity of a zero $r$ if $(x-r)^k$ is a factor of $f(x)$?
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Multiplicity is $k$. The power of the factor $(x-r)$ in the factorization.
Multiplicity is $k$. The power of the factor $(x-r)$ in the factorization.
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What is the Factor Theorem stated using $f(r)$ and $(x-r)$?
What is the Factor Theorem stated using $f(r)$ and $(x-r)$?
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$(x-r)$ is a factor of $f(x)$ iff $f(r)=0$. The fundamental connection between factors and zeros.
$(x-r)$ is a factor of $f(x)$ iff $f(r)=0$. The fundamental connection between factors and zeros.
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What is the end behavior if degree is even and leading coefficient is positive?
What is the end behavior if degree is even and leading coefficient is positive?
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As $x\to\pm\infty$, $f(x)\to\infty$. Even degree with positive lead: both ends go up.
As $x\to\pm\infty$, $f(x)\to\infty$. Even degree with positive lead: both ends go up.
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What does it mean if $(x-r)$ is a factor of $f(x)$?
What does it mean if $(x-r)$ is a factor of $f(x)$?
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$r$ is a zero of $f(x)$. Factor Theorem: $(x-r)$ is a factor iff $r$ is a zero.
$r$ is a zero of $f(x)$. Factor Theorem: $(x-r)$ is a factor iff $r$ is a zero.
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What is the x-intercept of the graph of $y=f(x)$ in terms of zeros?
What is the x-intercept of the graph of $y=f(x)$ in terms of zeros?
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Any point $(r,0)$ where $r$ is a zero of $f(x)$. Zeros create x-intercepts where the graph crosses the x-axis.
Any point $(r,0)$ where $r$ is a zero of $f(x)$. Zeros create x-intercepts where the graph crosses the x-axis.
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What does an odd multiplicity zero do to the graph at $x=r$?
What does an odd multiplicity zero do to the graph at $x=r$?
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The graph crosses the x-axis at $x=r$. Odd multiplicities cause the graph to pass through.
The graph crosses the x-axis at $x=r$. Odd multiplicities cause the graph to pass through.
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What is the end behavior if degree is even and leading coefficient is negative?
What is the end behavior if degree is even and leading coefficient is negative?
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As $x\to\pm\infty$, $f(x)\to-\infty$. Even degree with negative lead: both ends go down.
As $x\to\pm\infty$, $f(x)\to-\infty$. Even degree with negative lead: both ends go down.
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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 $f(r)=0$. When the function equals zero at that input value.
A value $r$ such that $f(r)=0$. When the function equals zero at that input value.
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Identify the zeros of $f(x)=(x-5)^2$ and state the multiplicity.
Identify the zeros of $f(x)=(x-5)^2$ and state the multiplicity.
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$x=5$ with multiplicity $2$. Only one distinct zero from the squared factor.
$x=5$ with multiplicity $2$. Only one distinct zero from the squared factor.
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Which end behavior matches $f(x)=2(x+1)^2(x-4)^2$?
Which end behavior matches $f(x)=2(x+1)^2(x-4)^2$?
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As $x\to\pm\infty$, $f(x)\to\infty$. Degree $4$ (even) with positive leading coefficient.
As $x\to\pm\infty$, $f(x)\to\infty$. Degree $4$ (even) with positive leading coefficient.
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Which end behavior matches $f(x)=-3(x-1)(x+2)(x-5)$?
Which end behavior matches $f(x)=-3(x-1)(x+2)(x-5)$?
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As $x\to-\infty$, $f(x)\to\infty$; as $x\to\infty$, $f(x)\to-\infty$. Degree $3$ (odd) with negative leading coefficient.
As $x\to-\infty$, $f(x)\to\infty$; as $x\to\infty$, $f(x)\to-\infty$. Degree $3$ (odd) with negative leading coefficient.
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What is the leading coefficient of $f(x)=-2(x-4)(x+1)^2$?
What is the leading coefficient of $f(x)=-2(x-4)(x+1)^2$?
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Leading coefficient $-2$. Coefficient of the highest degree term when expanded.
Leading coefficient $-2$. Coefficient of the highest degree term when expanded.
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What is the degree of $f(x)=(x-1)^2(x+3)$?
What is the degree of $f(x)=(x-1)^2(x+3)$?
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Degree $3$. Count the highest power when expanded.
Degree $3$. Count the highest power when expanded.
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Identify the zeros of $f(x)=(x^2-1)(x^2-9)$.
Identify the zeros of $f(x)=(x^2-1)(x^2-9)$.
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$x=\pm^1$ and $x=\pm^3$. Set each factor equal to zero and solve.
$x=\pm^1$ and $x=\pm^3$. Set each factor equal to zero and solve.
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Identify the zeros of $f(x)=x^4-16$ by factoring completely over reals.
Identify the zeros of $f(x)=x^4-16$ by factoring completely over reals.
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$x=2$ and $x=-2$. Factor as difference of squares twice: $(x^2-4)(x^2+4)=(x-2)(x+2)(x^2+4)$.
$x=2$ and $x=-2$. Factor as difference of squares twice: $(x^2-4)(x^2+4)=(x-2)(x+2)(x^2+4)$.
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Identify the zeros of $f(x)=x^3+2x^2-x-2$ given $f(x)=(x+2)(x+1)(x-1)$.
Identify the zeros of $f(x)=x^3+2x^2-x-2$ given $f(x)=(x+2)(x+1)(x-1)$.
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$x=-2$, $x=-1$, and $x=1$. Use the given factorization to find zeros.
$x=-2$, $x=-1$, and $x=1$. Use the given factorization to find zeros.
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Identify the zeros of $f(x)=x^3-9x$ by factoring.
Identify the zeros of $f(x)=x^3-9x$ by factoring.
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$x=0$, $x=3$, and $x=-3$. Factor out $x$: $x(x^2-9)=x(x-3)(x+3)=0$.
$x=0$, $x=3$, and $x=-3$. Factor out $x$: $x(x^2-9)=x(x-3)(x+3)=0$.
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Identify the zeros of $f(x)=x^3-4x^2$ by factoring.
Identify the zeros of $f(x)=x^3-4x^2$ by factoring.
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$x=0$ (mult. $2$) and $x=4$. Factor out $x^2$: $x^2(x-4)=0$.
$x=0$ (mult. $2$) and $x=4$. Factor out $x^2$: $x^2(x-4)=0$.
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Identify the zeros of $f(x)=x^2+7x+12$ by factoring.
Identify the zeros of $f(x)=x^2+7x+12$ by factoring.
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$x=-3$ and $x=-4$. Factor the quadratic: $(x+3)(x+4)=0$.
$x=-3$ and $x=-4$. Factor the quadratic: $(x+3)(x+4)=0$.
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Identify the zeros of $f(x)=x^2-2x-15$ by factoring.
Identify the zeros of $f(x)=x^2-2x-15$ by factoring.
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$x=5$ and $x=-3$. Factor the quadratic: $(x-5)(x+3)=0$.
$x=5$ and $x=-3$. Factor the quadratic: $(x-5)(x+3)=0$.
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Identify the zeros of $f(x)=2x^2-8x$ by factoring.
Identify the zeros of $f(x)=2x^2-8x$ by factoring.
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$x=0$ and $x=4$. Factor out $2x$: $2x(x-4)=0$.
$x=0$ and $x=4$. Factor out $2x$: $2x(x-4)=0$.
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Identify the zeros of $f(x)=x^2-6x+9$ by factoring.
Identify the zeros of $f(x)=x^2-6x+9$ by factoring.
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$x=3$ (multiplicity $2$). Perfect square trinomial: $(x-3)^2=0$.
$x=3$ (multiplicity $2$). Perfect square trinomial: $(x-3)^2=0$.
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Identify the zeros of $f(x)=x^2+5x$ by factoring.
Identify the zeros of $f(x)=x^2+5x$ by factoring.
Tap to reveal answer
$x=0$ and $x=-5$. Factor out common $x$: $x(x+5)=0$.
$x=0$ and $x=-5$. Factor out common $x$: $x(x+5)=0$.
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Identify the zeros of $f(x)=x^2-9$ by factoring.
Identify the zeros of $f(x)=x^2-9$ by factoring.
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$x=3$ and $x=-3$. Factor as difference of squares: $(x-3)(x+3)$.
$x=3$ and $x=-3$. Factor as difference of squares: $(x-3)(x+3)$.
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What is the y-intercept of $f(x)=-(x-1)^2(x+3)$?
What is the y-intercept of $f(x)=-(x-1)^2(x+3)$?
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$(0,-3)$. Substitute $x=0$: $f(0)=-(0-1)^2(0+3)=-3$.
$(0,-3)$. Substitute $x=0$: $f(0)=-(0-1)^2(0+3)=-3$.
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