Problems on Remainder Theorem

We will discuss here how to solve the problems on Remainder Theorem.

1. Find the remainder (without division) when 8x\(^{2}\) +5x + 1 is divisible by x - 10

Solution:

Here, f(x) = 8x\(^{2}\) + 5x + 1.

By remainder Theorem,

The remainder when f(x) is divided by x – 10 is f(10).


2. Find the remainder when x\(^{3}\) - ax\(^{2}\) + 6x - a is divisible by x - a.

Solution:

Here, f(x) = x\(^{3}\) - ax\(^{2}\) + 6x - a, divisor is (x - a)

Therefore, remainder = f(a) , [ Taking x = a from x - a = 0]

                                   = a\(^{3}\) - a ∙ a\(^{2}\) + 6 ∙ a - a

                                   = a\(^{3}\) -a\(^{3}\) + 6a - a

                                   = 5a.

3. Find the remainder (without division) when x\(^{2}\) +7x - 11 is divisible by 3x - 2

Solution:

Here, f(x) = x\(^{2}\) + 7x – 11 and 3x - 2 = 0 ⟹  x = \(\frac{2}{3}\)

By remainder Theorem,

The remainder when f(x) is divided by 3x - 2 is f(\(\frac{2}{3}\)).

Therefore, remainder = f(\(\frac{2}{3}\)) = (\(\frac{2}{3}\))\(^{2}\) + 7 ∙ (\(\frac{2}{3}\)) - 11

= \(\frac{4}{9}\) + \(\frac{14}{3}\) - 11

= -\(\frac{53}{9}\)



4. Check whether 7 + 3x is a factor of 3x\(^{3}\) + 7x.

Solution:

Here f(x) = 3x\(^{3}\) + 7x and divisor is 7 + 3x

Therefore, remainder = f(-\(\frac{7}{3}\)), [Taking x = -\(\frac{7}{3}\) from 7 + 3x = 0]

                                   = 3 ∙ (-\(\frac{7}{3}\))\(^{3}\) + 7(-\(\frac{7}{3}\))

                                   = -3 × \(\frac{343}{27}\) - \(\frac{49}{3}\)

                                   = \(\frac{-343 - 147}{9}\)

                                   = \(\frac{-490}{9}\)

                                   ≠ 0

Hence, 7 + 3x is not a factor of f(x) = 3x\(^{3}\) + 7x.



5. Find the remainder (without division) when 4x\(^{3}\) - 3x\(^{2}\) + 2x - 4 is divisible by x + 2

Solution:

Here, f(x) = 4x\(^{3}\) - 3x\(^{2}\) + 2x - 4 and x + 2 = 0 ⟹  x = -2

By remainder Theorem,

The remainder when f(x) is divided by x + 2 is f(-2).

Therefore, remainder = f(-2) = 4(-2)\(^{3}\) - 3 ∙ (-2)\(^{2}\) + 2 ∙ (-2) - 4

= - 32 - 12 - 4 - 4

= -52



6. Check whether the polynomial: f(x) = 4x\(^{3}\) + 4x\(^{2}\) - x - 1 is a multiple of 2x + 1.

Solution:

f(x) = 4x\(^{3}\) + 4x\(^{2}\) - x - 1 and divisor is 2x + 1

Therefore, remainder = f(-\(\frac{1}{2}\)), [Taking x = \(\frac{-1}{2}\) from 2x + 1 = 0]

                                   = 4 ∙ (-\(\frac{1}{2}\))\(^{3}\) + 4(-\(\frac{1}{2}\))\(^{2}\) - (-\(\frac{1}{2}\)) -1

                                   = -\(\frac{1}{2}\) + 1 + \(\frac{1}{2}\) - 1

                                   = 0

Since the remainder is zero ⟹ (2x + 1) is a factor of f(x). That is f(x) is a multiple of (2x + 1).

● Factorization




10th Grade Math

From Problems on Remainder Theorem to HOME


New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.



Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



Share this page: What’s this?

Recent Articles

  1. Months of the Year | List of 12 Months of the Year |Jan, Feb, Mar, Apr

    Dec 01, 23 01:16 AM

    Months of the Year
    There are 12 months in a year. The months are January, February, march, April, May, June, July, August, September, October, November and December. The year begins with the January month. December is t…

    Read More

  2. Days of the Week | 7 Days of the Week | What are the Seven Days?

    Nov 30, 23 10:59 PM

    Days of the Weeks
    We know that, seven days of a week are Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday. A day has 24 hours. There are 52 weeks in a year. Fill in the missing dates and answer the questi…

    Read More

  3. Types of Lines |Straight Lines|Curved Lines|Horizontal Lines| Vertical

    Nov 30, 23 01:08 PM

    Types of Lines
    What are the different types of lines? There are two different kinds of lines. (i) Straight line and (ii) Curved line. There are three different types of straight lines. (i) Horizontal lines, (ii) Ver…

    Read More