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Remainder and Factor Theorem

The Remainder Theorem and Factor Theorem are essential tools in polynomial algebra. They allow us to evaluate polynomials quickly and determine whether a given binomial is a factor of the polynomial, without performing full division.

Remainder Theorem:

If a polynomial $f(x)$ is divided by a linear divisor of the form $(x - c)$, the remainder is simply:

$$ \text{Remainder} = f(c) $$

This means you can find the remainder by directly substituting $x = c$ into the polynomial, avoiding long division.

Factor Theorem:

A special case of the Remainder Theorem: If $f(c) = 0$, then $(x - c)$ is a factor of $f(x)$.

$$ f(c) = 0 \quad \Rightarrow \quad (x - c) \text{ is a factor of } f(x) $$

This theorem is commonly used to find roots or to start factoring higher-degree polynomials.

How They Work Together:

These theorems simplify root-finding and polynomial factoring, especially for higher-order polynomials in engineering equations or signal analysis.

Concept Concept Concept Concept Concept Concept Concept Concept Concept

Problem:

What is the remainder when $9-3x+5x^2+6x^3$ is divided by $3x-2$?

Remainder and Factor Theorem | Algebra – Problem 1: – Diagram Remainder and Factor Theorem | Algebra – Problem 1: – Diagram Remainder and Factor Theorem | Algebra – Problem 1: – Diagram

Using all methods: Long Division, Synthetic Division, and Remainder Theorem

Remainder and Factor Theorem | Algebra – Problem 1: – Diagram Remainder and Factor Theorem | Algebra – Problem 1: – Diagram Remainder and Factor Theorem | Algebra – Problem 1: – Diagram Remainder and Factor Theorem | Algebra – Problem 1: – Diagram

Problem:

When the expression $x^4+ax^3+5x^2+bx+6$ is divided by $(x-2)$, the remainder is 16. When it is divided by $(x+1)$, the remainder is 10. Find the value of the constant b.

Remainder and Factor Theorem | Algebra – Problem 2: – Diagram Remainder and Factor Theorem | Algebra – Problem 2: – Diagram Remainder and Factor Theorem | Algebra – Problem 2: – Diagram

Given $f(x)=x^4+ax^3+5x^2+bx+6$. When divided by $(x-2)$ the remainder is $16$ $\Rightarrow$ $f(2)=16$. When divided by $(x+1)$ the remainder is $10$ $\Rightarrow$ $f(-1)=10$.

  1. $f(2)=2^4+a(2)^3+5(2)^2+b(2)+6=16+8a+20+2b+6=42+8a+2b$. Set $42+8a+2b=16 \Rightarrow 8a+2b=-26 \Rightarrow 4a+b=-13$.
  2. $f(-1)=(-1)^4+a(-1)^3+5(-1)^2+b(-1)+6=1-a+5-b+6=12-a-b$. Set $12-a-b=10 \Rightarrow a+b=2$.
  3. Solve the system: $\begin{cases}4a+b=-13\\ a+b=2\end{cases}\Rightarrow 3a=-15\Rightarrow a=-5,\ b=7$.

Therefore, $\boxed{b=7}$.

Check: $f(2)=42+8(-5)+2(7)=16$, $f(-1)=12-(-5)-7=10$.

Remainder and Factor Theorem | Algebra – Problem 2: – Diagram Remainder and Factor Theorem | Algebra – Problem 2: – Diagram Remainder and Factor Theorem | Algebra – Problem 2: – Diagram Remainder and Factor Theorem | Algebra – Problem 2: – Diagram

Problem:

Find k so that $2x^2-kx-9$ has x-k as a factor.

Remainder and Factor Theorem | Algebra – Problem 3: – Diagram Remainder and Factor Theorem | Algebra – Problem 3: – Diagram Remainder and Factor Theorem | Algebra – Problem 3: – Diagram

  1. If $x-k$ is a factor, then by the Factor Theorem we must have $f(k)=0$ for $f(x)=2x^2-kx-9$. Note that "the divisor is a factor" essentially means we do not have a remainder.
  2. Substitute $x=k$: $$f(k)=2k^2-k(k)-9=2k^2-k^2-9=k^2-9.$$
  3. Set equal to zero: $$k^2-9=0 \;\;\Rightarrow\;\; k^2=9.$$
  4. Solve for $k$: $$k=\pm 3.$$

Final Answer: $\boxed{k=3}$ or $\boxed{k=-3}$

Remainder and Factor Theorem | Algebra – Problem 3: – Diagram Remainder and Factor Theorem | Algebra – Problem 3: – Diagram Remainder and Factor Theorem | Algebra – Problem 3: – Diagram Remainder and Factor Theorem | Algebra – Problem 3: – Diagram

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Remainder and Factor Theorem | Algebra – Problem 4: – Diagram Remainder and Factor Theorem | Algebra – Problem 4: – Diagram Remainder and Factor Theorem | Algebra – Problem 4: – Diagram

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Remainder and Factor Theorem | Algebra – Problem 4: – Diagram Remainder and Factor Theorem | Algebra – Problem 4: – Diagram Remainder and Factor Theorem | Algebra – Problem 4: – Diagram Remainder and Factor Theorem | Algebra – Problem 4: – Diagram

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Remainder and Factor Theorem | Algebra – Problem 5: – Diagram Remainder and Factor Theorem | Algebra – Problem 5: – Diagram Remainder and Factor Theorem | Algebra – Problem 5: – Diagram

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Remainder and Factor Theorem | Algebra – Problem 5: – Diagram Remainder and Factor Theorem | Algebra – Problem 5: – Diagram Remainder and Factor Theorem | Algebra – Problem 5: – Diagram Remainder and Factor Theorem | Algebra – Problem 5: – Diagram

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Remainder and Factor Theorem | Algebra – Problem 6: – Diagram Remainder and Factor Theorem | Algebra – Problem 6: – Diagram Remainder and Factor Theorem | Algebra – Problem 6: – Diagram

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Remainder and Factor Theorem | Algebra – Problem 6: – Diagram Remainder and Factor Theorem | Algebra – Problem 6: – Diagram Remainder and Factor Theorem | Algebra – Problem 6: – Diagram Remainder and Factor Theorem | Algebra – Problem 6: – Diagram

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Remainder and Factor Theorem | Algebra – Problem 7: – Diagram Remainder and Factor Theorem | Algebra – Problem 7: – Diagram Remainder and Factor Theorem | Algebra – Problem 7: – Diagram Remainder and Factor Theorem | Algebra – Problem 7: – Diagram

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Remainder and Factor Theorem | Algebra – Problem 8: – Diagram Remainder and Factor Theorem | Algebra – Problem 8: – Diagram Remainder and Factor Theorem | Algebra – Problem 8: – Diagram

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Remainder and Factor Theorem | Algebra – Problem 8: – Diagram Remainder and Factor Theorem | Algebra – Problem 8: – Diagram Remainder and Factor Theorem | Algebra – Problem 8: – Diagram Remainder and Factor Theorem | Algebra – Problem 8: – Diagram
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Exam Generator Problems

Additional board-style practice items for this topic.

Question Bank: q257

MSTE - Algebra / Remainder and Factor Theorem / Engr. Janclyde Espinosa (Clidez)

The polynomial x3 + 4x2 - 3x + 8 is divided by x - 5. What is the remainder?

Answer:

  1. 218
  2. 281
  3. 812
  4. 182
By the Remainder Theorem, the remainder when $f(x)$ is divided by $x-5$ is $f(5)$.
$f(5)=5^3+4(5^2)-3(5)+8$
$=125+100-15+8$
$\boxed{218}$

Question Bank: q258

MSTE - Algebra / Remainder and Factor Theorem / Engr. Janclyde Espinosa (Clidez)

Solve for x from the following equations:
xy=12
yz=20
zx=15

Answer:

  1. 3
  2. 4
  3. 5
  4. 2
Multiply and divide the given products:
$x^2=\frac{(xy)(xz)}{yz}=\frac{12(15)}{20}=9$
$\boxed{x=3}$

Question Bank: q391

MSTE - Algebra / Remainder and Factor Theorem / Engr. Janclyde Espinosa (Clidez)

When x3+kx2-4x+2 is divided by x+2, the remainder is 26. What is the value of k?

Answer:

  1. 6
  2. 4
  3. 2
  4. 3
By the Remainder Theorem, substituting $x=-2$ gives the remainder 26:
$(-2)^3+k(-2)^2-4(-2)+2=26$
$-8+4k+8+2=26$
$4k=24$
$\boxed{k=6}$

Question Bank: q392

MSTE - Algebra / Remainder and Factor Theorem / Engr. Janclyde Espinosa (Clidez)

If f(x) = ax3+5x2-23x-6 and 4x+1 is a factor, determine the value of a.

Answer:

  1. 4
  2. 3
  3. 6
  4. 2
If $4x+1$ is a factor, then $x=-1/4$ is a root:
$a(-1/4)^3+5(-1/4)^2-23(-1/4)-6=0$
$-\frac{a}{64}+\frac{5}{16}+\frac{23}{4}-6=0$
$-\frac{a}{64}+\frac{1}{16}=0$
$\boxed{a=4}$

Question Bank: q690

MSTE - Algebra / Remainder and Factor Theorem / Engr. Janclyde Espinosa (Clidez)

The polynomial:
f(x)=ax3-10x2-3x+b has a factor of x+5. When f(x) is divided by x-2, the remainder is -77. Find the values of a and b.

  1. a = −2, b = −15
  2. a = 2, b = 15
  3. a = −2, b = 1
  4. a = 2, b = −15
Use the factor and remainder conditions. From $f(-5)=0$:
$-125a-250+15+b=0$, so $b=125a+235$. From $f(2)=-77$:
$8a-40-6+b=-77$, so $b=-31-8a$. Solving gives:
$a=-2$, $b=-15$
$\boxed{a=-2,\ b=-15}$

Question Bank: t67

MSTE - Algebra / Algebra Fundamentals / Gemini mapped Chapter 1 to 3

If a negative number is divided by a positive number, which of the following must be the quotient?

  1. negative
  2. positive
  3. integer
  4. fraction
A negative number divided by a positive number always gives a negative quotient.
Example: $-6 \div 3 = -2$
$\boxed{\text{negative}}$

Question Bank: t76

MSTE - Algebra / Algebra Fundamentals / Gemini mapped Chapter 1 to 3

A line is divided into 10 equal parts. If the measure of each part is a prime integer, what is the possible length of the line?

  1. 150
  2. 90
  3. 210
  4. 230
The line is divided into 10 equal parts, and each part must be a prime integer. Therefore the total length must be $10p$, where $p$ is prime.
Check the choices by dividing by 10:
$230 \div 10 = 23$, and 23 is prime.
$\boxed{230}$

Question Bank: t93

MSTE - Algebra / Algebra Fundamentals / Gemini mapped Chapter 1 to 3

If $x/y$ gives a remainder of 0 and $z/y$ gives a remainder of 3, what is the remainder of $xz/y$?

  1. 0
  2. 3
  3. 2
  4. 1
If $x/y$ has remainder 0, then $x$ is divisible by $y$. Therefore $x = ky$ for some integer $k$.
Then $xz = (ky)z = kzy$, which is also divisible by $y$.
So the remainder is:
$\boxed{0}$

Question Bank: t101

MSTE - Algebra / Algebra Fundamentals / Gemini mapped Chapter 1 to 3

If $(x^2 + 9x + 14)/(x^2 - 49)$ is divided by $(3x + 6)/(x^2 + x - 56)$, the quotient is:

  1. $x + 8$
  2. $x + 3$
  3. $(x + 8)/3$
  4. $(x + 3)/8$
$\frac{x^2+9x+14}{x^2-49} \div \frac{3x+6}{x^2+x-56}$
Factor: $x^2+9x+14=(x+2)(x+7)$; $x^2-49=(x+7)(x-7)$
$3x+6=3(x+2)$; $x^2+x-56=(x+8)(x-7)$
$= \frac{(x+2)(x+7)}{(x+7)(x-7)} \times \frac{(x+8)(x-7)}{3(x+2)} = \frac{x+8}{3}$
$\boxed{= \frac{x+8}{3}}$

Question Bank: t102

MSTE - Algebra / Algebra Fundamentals / Gemini mapped Chapter 1 to 3

Which of the following is a factor of $x^3 - 3x^2 + 6x - 4$?

  1. $x^2 + 2x - 4$
  2. $x^2 - 2x - 4$
  3. $x^2 + 2x + 4$
  4. $x^2 - 2x + 4$
$x^3-3x^2+6x-4$. Try $x=1$: $1-3+6-4=0$ → $(x-1)$ is a factor.
Divide: $x^3-3x^2+6x-4 = (x-1)(x^2-2x+4)$
$\boxed{x^2-2x+4}$

Question Bank: t104

MSTE - Algebra / Algebra Fundamentals / Gemini mapped Chapter 1 to 3

When $f(x) = x^4 + ax^3 + 7x^2 + bx + 6$ is divided by $(x - 2)$ the remainder is 16 and when divided by $(x + 1)$ the remainder is 10. What is the value of $a$?

  1. -7
  2. 11
  3. -5
  4. 3
By the Remainder Theorem, $f(2)=16$ and $f(-1)=10$.
$f(2)=16+8a+28+2b+6=16 \Rightarrow 4a+b=-17$
$f(-1)=1-a+7-b+6=10 \Rightarrow a+b=4$
Subtract: $(4a+b)-(a+b)=-17-4 \Rightarrow 3a=-21$
$\boxed{a=-7}$

Question Bank: t1358

MSTE - Algebra / Additional Algebra Problems / Gemini mapped Chapter 7 to 10

Factor x^4 - y^2 + y - x^2 as possible.

  1. (x^2 - y) (x^2 + y)
  2. (x^2 + y) (x^2 + y - 1)
  3. (x^2 + y) (x^2 - y - 1)
  4. (x^2 - y) (x^2 + y - 1)

Solution pending in psadquestions/t1358.json.

Question Bank: t1367

MSTE - Algebra / Additional Algebra Problems / Gemini mapped Chapter 7 to 10

Given that f(x) = (x - 4)(x + 3) + 9, when f(x) is divided by (x - k) the remainder is k. Find the positive value of k.

  1. 4
  2. 5
  3. 3
  4. 2

Solution pending in psadquestions/t1367.json.