Polynomial Equations - PSAT Math
Card 1 of 30
Factor completely: $x^2-16$.
Factor completely: $x^2-16$.
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$(x-4)(x+4)$. Difference of squares: $x^2-4^2=(x-4)(x+4)$.
$(x-4)(x+4)$. Difference of squares: $x^2-4^2=(x-4)(x+4)$.
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Find the sum of the solutions of $x^2-7x+10=0$ without solving fully.
Find the sum of the solutions of $x^2-7x+10=0$ without solving fully.
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$7$. By Vieta's formulas, sum of roots equals negative coefficient of $x$ divided by leading coefficient.
$7$. By Vieta's formulas, sum of roots equals negative coefficient of $x$ divided by leading coefficient.
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What is the maximum number of real solutions a degree $n$ polynomial equation can have?
What is the maximum number of real solutions a degree $n$ polynomial equation can have?
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At most $n$ real solutions. By the Fundamental Theorem of Algebra, counting multiplicities.
At most $n$ real solutions. By the Fundamental Theorem of Algebra, counting multiplicities.
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Factor completely: $x^2+6x+9$.
Factor completely: $x^2+6x+9$.
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$(x+3)^2$. Recognizes as $(x)^2+2(x)(3)+(3)^2$, a perfect square trinomial.
$(x+3)^2$. Recognizes as $(x)^2+2(x)(3)+(3)^2$, a perfect square trinomial.
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What is the factored form of the perfect square trinomial $a^2+2ab+b^2$?
What is the factored form of the perfect square trinomial $a^2+2ab+b^2$?
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$(a+b)^2$. Perfect square trinomial: first term squared plus twice the product plus last squared.
$(a+b)^2$. Perfect square trinomial: first term squared plus twice the product plus last squared.
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What is the factored form of the difference of squares $a^2-b^2$?
What is the factored form of the difference of squares $a^2-b^2$?
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$(a-b)(a+b)$. This is the difference of squares factorization pattern.
$(a-b)(a+b)$. This is the difference of squares factorization pattern.
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How many real solutions does $x^2+4x+5=0$ have?
How many real solutions does $x^2+4x+5=0$ have?
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$0$ real solutions. Discriminant $=16-20=-4<0$, so no real solutions exist.
$0$ real solutions. Discriminant $=16-20=-4<0$, so no real solutions exist.
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Identify the zeros of $f(x)=(x-2)(x+5)$.
Identify the zeros of $f(x)=(x-2)(x+5)$.
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$x=2$ and $x=-5$. Set each factor equal to zero: $(x-2)=0$ and $(x+5)=0$.
$x=2$ and $x=-5$. Set each factor equal to zero: $(x-2)=0$ and $(x+5)=0$.
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What is the Factor Theorem statement relating $f(r)$ to the factor $(x-r)$?
What is the Factor Theorem statement relating $f(r)$ to the factor $(x-r)$?
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$(x-r)$ is a factor of $f(x)$ iff $f(r)=0$. A polynomial has factor $(x-r)$ exactly when $r$ is a root.
$(x-r)$ is a factor of $f(x)$ iff $f(r)=0$. A polynomial has factor $(x-r)$ exactly when $r$ is a root.
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What does the Remainder Theorem say is the remainder when dividing $f(x)$ by $(x-c)$?
What does the Remainder Theorem say is the remainder when dividing $f(x)$ by $(x-c)$?
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Remainder $=f(c)$. When dividing $f(x)$ by $(x-c)$, the remainder equals $f(c)$.
Remainder $=f(c)$. When dividing $f(x)$ by $(x-c)$, the remainder equals $f(c)$.
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What is the standard form of a polynomial equation in $x$ of degree $n$?
What is the standard form of a polynomial equation in $x$ of degree $n$?
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$a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0=0$, $a_n\neq 0$. Standard form has terms in descending degree order with leading coefficient non-zero.
$a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0=0$, $a_n\neq 0$. Standard form has terms in descending degree order with leading coefficient non-zero.
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Solve the polynomial equation $(x-4)(x+1)=0$.
Solve the polynomial equation $(x-4)(x+1)=0$.
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$x=4$ or $x=-1$. Set each factor to zero: $x-4=0$ or $x+1=0$.
$x=4$ or $x=-1$. Set each factor to zero: $x-4=0$ or $x+1=0$.
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Identify the number of real solutions to $x^2-6x+9=0$.
Identify the number of real solutions to $x^2-6x+9=0$.
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$1$ real solution. $(x-3)^2=0$ has repeated root.
$1$ real solution. $(x-3)^2=0$ has repeated root.
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Identify the solutions of $x^2+2x-3=0$.
Identify the solutions of $x^2+2x-3=0$.
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$x=1$ and $x=-3$. Factor as $(x-1)(x+3)=0$.
$x=1$ and $x=-3$. Factor as $(x-1)(x+3)=0$.
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What is the definition of a polynomial equation in one variable $x$?
What is the definition of a polynomial equation in one variable $x$?
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An equation setting a polynomial in $x$ equal to $0$. Sets polynomial to zero to find roots.
An equation setting a polynomial in $x$ equal to $0$. Sets polynomial to zero to find roots.
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Identify the solutions of $(x-4)(x+1)=0$.
Identify the solutions of $(x-4)(x+1)=0$.
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$x=4$ and $x=-1$. Apply zero product property.
$x=4$ and $x=-1$. Apply zero product property.
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What is the remainder when $f(x)=x^3-2x+5$ is divided by $(x-2)$?
What is the remainder when $f(x)=x^3-2x+5$ is divided by $(x-2)$?
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$f(2)=9$. By Remainder Theorem, substitute $x=2$.
$f(2)=9$. By Remainder Theorem, substitute $x=2$.
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Which value is a root of $f(x)=x^3-4x^2-x+4$: $x=1$ or $x=2$?
Which value is a root of $f(x)=x^3-4x^2-x+4$: $x=1$ or $x=2$?
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$x=1$. $f(1)=1-4-1+4=0$, so $x=1$ is a root.
$x=1$. $f(1)=1-4-1+4=0$, so $x=1$ is a root.
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Identify the number of real solutions to $x^2+4x+5=0$.
Identify the number of real solutions to $x^2+4x+5=0$.
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$0$ real solutions. Discriminant $16-20=-4<0$.
$0$ real solutions. Discriminant $16-20=-4<0$.
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What is the factorization pattern for a perfect square trinomial $a^2+2ab+b^2$?
What is the factorization pattern for a perfect square trinomial $a^2+2ab+b^2$?
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$a^2+2ab+b^2=(a+b)^2$. Perfect square trinomial factors to squared binomial.
$a^2+2ab+b^2=(a+b)^2$. Perfect square trinomial factors to squared binomial.
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What is the discriminant of $2x^2-3x+5=0$?
What is the discriminant of $2x^2-3x+5=0$?
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$-31$. Calculate $b^2-4ac = (-3)^2-4(2)(5) = 9-40$.
$-31$. Calculate $b^2-4ac = (-3)^2-4(2)(5) = 9-40$.
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What is the degree of the polynomial $7x^4-3x^2+9$?
What is the degree of the polynomial $7x^4-3x^2+9$?
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$4$. The degree is the highest power of the variable.
$4$. The degree is the highest power of the variable.
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What is the leading coefficient of $-5x^3+2x-1$?
What is the leading coefficient of $-5x^3+2x-1$?
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$-5$. The leading coefficient is the coefficient of the highest degree term.
$-5$. The leading coefficient is the coefficient of the highest degree term.
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What is the standard form of a polynomial in descending powers of $x$?
What is the standard form of a polynomial in descending powers of $x$?
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$a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$. Terms arranged from highest to lowest degree.
$a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$. Terms arranged from highest to lowest degree.
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What is the zero-product property used to solve $(x-3)(x+5)=0$?
What is the zero-product property used to solve $(x-3)(x+5)=0$?
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If $ab=0$, then $a=0$ or $b=0$. A product equals zero only if at least one factor is zero.
If $ab=0$, then $a=0$ or $b=0$. A product equals zero only if at least one factor is zero.
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What are the solutions of $(x-3)(x+5)=0$?
What are the solutions of $(x-3)(x+5)=0$?
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$x=3$ or $x=-5$. Apply zero-product property: set each factor to zero.
$x=3$ or $x=-5$. Apply zero-product property: set each factor to zero.
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What is the factorization pattern for a difference of squares $a^2-b^2$?
What is the factorization pattern for a difference of squares $a^2-b^2$?
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$a^2-b^2=(a-b)(a+b)$. Difference of squares factors into sum times difference.
$a^2-b^2=(a-b)(a+b)$. Difference of squares factors into sum times difference.
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What is the complete factorization of $x^2-49$ over the integers?
What is the complete factorization of $x^2-49$ over the integers?
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$(x-7)(x+7)$. $49 = 7^2$, so apply difference of squares pattern.
$(x-7)(x+7)$. $49 = 7^2$, so apply difference of squares pattern.
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What is the factorization pattern for a perfect square trinomial $a^2-2ab+b^2$?
What is the factorization pattern for a perfect square trinomial $a^2-2ab+b^2$?
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$a^2-2ab+b^2=(a-b)^2$. Perfect square with subtraction factors to squared difference.
$a^2-2ab+b^2=(a-b)^2$. Perfect square with subtraction factors to squared difference.
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What is the factorization of $x^2-10x+25$?
What is the factorization of $x^2-10x+25$?
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$(x-5)^2$. Recognize pattern: $(-5)^2 = 25$ and $2(-5) = -10$.
$(x-5)^2$. Recognize pattern: $(-5)^2 = 25$ and $2(-5) = -10$.
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