Applying the Remainder Theorem - Algebra 2
Card 1 of 30
If $p(-4)=12$, what is the remainder when dividing $p(x)$ by $x+4$?
If $p(-4)=12$, what is the remainder when dividing $p(x)$ by $x+4$?
Tap to reveal answer
Remainder $=12$. By the Remainder Theorem, the remainder equals $p(-4)$.
Remainder $=12$. By the Remainder Theorem, the remainder equals $p(-4)$.
← Didn't Know|Knew It →
Find the value of $k$ so that $x-2$ is a factor of $p(x)=x^2+kx-6$.
Find the value of $k$ so that $x-2$ is a factor of $p(x)=x^2+kx-6$.
Tap to reveal answer
$k=1$. Set $p(2)=0$: $4+2k-6=0$, so $k=1$.
$k=1$. Set $p(2)=0$: $4+2k-6=0$, so $k=1$.
← Didn't Know|Knew It →
Find the value of $k$ so that $x+3$ is a factor of $p(x)=2x^2+kx-9$.
Find the value of $k$ so that $x+3$ is a factor of $p(x)=2x^2+kx-9$.
Tap to reveal answer
$k=3$. Set $p(-3)=0$: $18-3k-9=0$, so $k=3$.
$k=3$. Set $p(-3)=0$: $18-3k-9=0$, so $k=3$.
← Didn't Know|Knew It →
Find the value of $k$ so that $x-1$ is a factor of $p(x)=x^3-4x^2+kx+3$.
Find the value of $k$ so that $x-1$ is a factor of $p(x)=x^3-4x^2+kx+3$.
Tap to reveal answer
$k=0$. Set $p(1)=0$: $1-4+k+3=0$, so $k=0$.
$k=0$. Set $p(1)=0$: $1-4+k+3=0$, so $k=0$.
← Didn't Know|Knew It →
What is the remainder when dividing $p(x)=x^3+2x^2-x-2$ by $x-1$?
What is the remainder when dividing $p(x)=x^3+2x^2-x-2$ by $x-1$?
Tap to reveal answer
$p(1)=0$. Calculate $p(1)=1^3+2(1)^2-1-2=1+2-1-2=0$.
$p(1)=0$. Calculate $p(1)=1^3+2(1)^2-1-2=1+2-1-2=0$.
← Didn't Know|Knew It →
What is the remainder when dividing $p(x)=x^3+2x^2-x-2$ by $x+2$?
What is the remainder when dividing $p(x)=x^3+2x^2-x-2$ by $x+2$?
Tap to reveal answer
$p(-2)=0$. Calculate $p(-2)=(-2)^3+2(-2)^2-(-2)-2=-8+8+2-2=0$.
$p(-2)=0$. Calculate $p(-2)=(-2)^3+2(-2)^2-(-2)-2=-8+8+2-2=0$.
← Didn't Know|Knew It →
What is the remainder when dividing $p(x)=x^3-2x^2+x-2$ by $x-2$?
What is the remainder when dividing $p(x)=x^3-2x^2+x-2$ by $x-2$?
Tap to reveal answer
$p(2)=0$. Calculate $p(2)=8-8+2-2=0$.
$p(2)=0$. Calculate $p(2)=8-8+2-2=0$.
← Didn't Know|Knew It →
What is the remainder when dividing $p(x)=x^3-2x^2+x-2$ by $x+1$?
What is the remainder when dividing $p(x)=x^3-2x^2+x-2$ by $x+1$?
Tap to reveal answer
$p(-1)=-6$. Calculate $p(-1)=-1-2-1-2=-6$.
$p(-1)=-6$. Calculate $p(-1)=-1-2-1-2=-6$.
← Didn't Know|Knew It →
Which statement is true if the remainder on division by $x-4$ is $0$?
Which statement is true if the remainder on division by $x-4$ is $0$?
Tap to reveal answer
$(x-4)$ is a factor of $p(x)$. A zero remainder means the divisor is a factor.
$(x-4)$ is a factor of $p(x)$. A zero remainder means the divisor is a factor.
← Didn't Know|Knew It →
What does the Remainder Theorem state for the remainder when dividing $p(x)$ by $x-a$?
What does the Remainder Theorem state for the remainder when dividing $p(x)$ by $x-a$?
Tap to reveal answer
Remainder $=p(a)$. This is the core statement of the Remainder Theorem.
Remainder $=p(a)$. This is the core statement of the Remainder Theorem.
← Didn't Know|Knew It →
What is the remainder when dividing $p(x)$ by $x-3$ in terms of $p$?
What is the remainder when dividing $p(x)$ by $x-3$ in terms of $p$?
Tap to reveal answer
Remainder $=p(3)$. By the Remainder Theorem, substitute $a=3$.
Remainder $=p(3)$. By the Remainder Theorem, substitute $a=3$.
← Didn't Know|Knew It →
What is the remainder when dividing $p(x)$ by $x+5$ in terms of $p$?
What is the remainder when dividing $p(x)$ by $x+5$ in terms of $p$?
Tap to reveal answer
Remainder $=p(-5)$. Since $x+5=x-(-5)$, we have $a=-5$.
Remainder $=p(-5)$. Since $x+5=x-(-5)$, we have $a=-5$.
← Didn't Know|Knew It →
What does the Factor Theorem state using $p(a)$ and the factor $(x-a)$?
What does the Factor Theorem state using $p(a)$ and the factor $(x-a)$?
Tap to reveal answer
$p(a)=0\iff (x-a)$ is a factor of $p(x)$. The Factor Theorem combines remainder and factorization.
$p(a)=0\iff (x-a)$ is a factor of $p(x)$. The Factor Theorem combines remainder and factorization.
← Didn't Know|Knew It →
Which condition guarantees that $(x-a)$ is a factor of $p(x)$: $p(a)=0$ or $p(a)\neq 0$?
Which condition guarantees that $(x-a)$ is a factor of $p(x)$: $p(a)=0$ or $p(a)\neq 0$?
Tap to reveal answer
$p(a)=0$. When $p(a)=0$, the remainder is zero, making $(x-a)$ a factor.
$p(a)=0$. When $p(a)=0$, the remainder is zero, making $(x-a)$ a factor.
← Didn't Know|Knew It →
If $(x-a)$ is a factor of $p(x)$, what must the remainder be when dividing by $(x-a)$?
If $(x-a)$ is a factor of $p(x)$, what must the remainder be when dividing by $(x-a)$?
Tap to reveal answer
Remainder $=0$. If $(x-a)$ is a factor, then $p(a)=0$.
Remainder $=0$. If $(x-a)$ is a factor, then $p(a)=0$.
← Didn't Know|Knew It →
Identify the value of $a$ used in the Remainder Theorem when the divisor is $x-7$.
Identify the value of $a$ used in the Remainder Theorem when the divisor is $x-7$.
Tap to reveal answer
$a=7$. For divisor $x-7$, the value $a=7$.
$a=7$. For divisor $x-7$, the value $a=7$.
← Didn't Know|Knew It →
Identify the value of $a$ used in the Remainder Theorem when the divisor is $x+7$.
Identify the value of $a$ used in the Remainder Theorem when the divisor is $x+7$.
Tap to reveal answer
$a=-7$. For divisor $x+7=x-(-7)$, the value $a=-7$.
$a=-7$. For divisor $x+7=x-(-7)$, the value $a=-7$.
← Didn't Know|Knew It →
What is the remainder when dividing $p(x)=x^3-4x+1$ by $x-2$?
What is the remainder when dividing $p(x)=x^3-4x+1$ by $x-2$?
Tap to reveal answer
$p(2)=1$. Calculate $p(2)=2^3-4(2)+1=8-8+1=1$.
$p(2)=1$. Calculate $p(2)=2^3-4(2)+1=8-8+1=1$.
← Didn't Know|Knew It →
What is the remainder when dividing $p(x)=x^2+3x-10$ by $x-2$?
What is the remainder when dividing $p(x)=x^2+3x-10$ by $x-2$?
Tap to reveal answer
$p(2)=0$. Calculate $p(2)=2^2+3(2)-10=4+6-10=0$.
$p(2)=0$. Calculate $p(2)=2^2+3(2)-10=4+6-10=0$.
← Didn't Know|Knew It →
What is the remainder when dividing $p(x)=x^2+3x-10$ by $x+5$?
What is the remainder when dividing $p(x)=x^2+3x-10$ by $x+5$?
Tap to reveal answer
$p(-5)=0$. Calculate $p(-5)=(-5)^2+3(-5)-10=25-15-10=0$.
$p(-5)=0$. Calculate $p(-5)=(-5)^2+3(-5)-10=25-15-10=0$.
← Didn't Know|Knew It →
What is the remainder when dividing $p(x)=2x^3-x^2+5$ by $x-1$?
What is the remainder when dividing $p(x)=2x^3-x^2+5$ by $x-1$?
Tap to reveal answer
$p(1)=6$. Calculate $p(1)=2(1)^3-(1)^2+5=2-1+5=6$.
$p(1)=6$. Calculate $p(1)=2(1)^3-(1)^2+5=2-1+5=6$.
← Didn't Know|Knew It →
What is the remainder when dividing $p(x)=2x^3-x^2+5$ by $x+1$?
What is the remainder when dividing $p(x)=2x^3-x^2+5$ by $x+1$?
Tap to reveal answer
$p(-1)=2$. Calculate $p(-1)=2(-1)^3-(-1)^2+5=-2-1+5=2$.
$p(-1)=2$. Calculate $p(-1)=2(-1)^3-(-1)^2+5=-2-1+5=2$.
← Didn't Know|Knew It →
What is the remainder when dividing $p(x)=x^4-16$ by $x-2$?
What is the remainder when dividing $p(x)=x^4-16$ by $x-2$?
Tap to reveal answer
$p(2)=0$. Calculate $p(2)=2^4-16=16-16=0$.
$p(2)=0$. Calculate $p(2)=2^4-16=16-16=0$.
← Didn't Know|Knew It →
What is the remainder when dividing $p(x)=x^4-16$ by $x+2$?
What is the remainder when dividing $p(x)=x^4-16$ by $x+2$?
Tap to reveal answer
$p(-2)=0$. Calculate $p(-2)=(-2)^4-16=16-16=0$.
$p(-2)=0$. Calculate $p(-2)=(-2)^4-16=16-16=0$.
← Didn't Know|Knew It →
If the remainder on division by $x-a$ is $r$, what is $p(a)$ equal to?
If the remainder on division by $x-a$ is $r$, what is $p(a)$ equal to?
Tap to reveal answer
$p(a)=r$. The remainder equals the polynomial evaluated at $a$.
$p(a)=r$. The remainder equals the polynomial evaluated at $a$.
← Didn't Know|Knew It →
If $p(6)=-3$, what is the remainder when dividing $p(x)$ by $x-6$?
If $p(6)=-3$, what is the remainder when dividing $p(x)$ by $x-6$?
Tap to reveal answer
Remainder $=-3$. By the Remainder Theorem, the remainder equals $p(6)$.
Remainder $=-3$. By the Remainder Theorem, the remainder equals $p(6)$.
← Didn't Know|Knew It →
If $p(0)=5$, what is the remainder when dividing $p(x)$ by $x$?
If $p(0)=5$, what is the remainder when dividing $p(x)$ by $x$?
Tap to reveal answer
Remainder $=5$. Dividing by $x$ means evaluating at $x=0$.
Remainder $=5$. Dividing by $x$ means evaluating at $x=0$.
← Didn't Know|Knew It →
What is the remainder when dividing $p(x)=3x^2-12x+9$ by $x-1$?
What is the remainder when dividing $p(x)=3x^2-12x+9$ by $x-1$?
Tap to reveal answer
$p(1)=0$. Calculate $p(1)=3(1)^2-12(1)+9=3-12+9=0$.
$p(1)=0$. Calculate $p(1)=3(1)^2-12(1)+9=3-12+9=0$.
← Didn't Know|Knew It →
Find the value of $k$ so that $x+2$ is a factor of $p(x)=x^3+kx^2-4x+8$.
Find the value of $k$ so that $x+2$ is a factor of $p(x)=x^3+kx^2-4x+8$.
Tap to reveal answer
$k=0$. Set $p(-2)=0$: $-8+4k+8+8=0$, so $k=0$.
$k=0$. Set $p(-2)=0$: $-8+4k+8+8=0$, so $k=0$.
← Didn't Know|Knew It →
What is the remainder when dividing $p(x)=3x^2-12x+9$ by $x-3$?
What is the remainder when dividing $p(x)=3x^2-12x+9$ by $x-3$?
Tap to reveal answer
$p(3)=0$. Calculate $p(3)=3(3)^2-12(3)+9=27-36+9=0$.
$p(3)=0$. Calculate $p(3)=3(3)^2-12(3)+9=27-36+9=0$.
← Didn't Know|Knew It →