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

• Polynomial
• Polynomial Equation and its Roots
  
• Division Algorithm
  
• Remainder Theorem
  
• Problems on Remainder Theorem
  
• Factors of a Polynomial
  
• Worksheet on Remainder Theorem
  
• Factor Theorem
  
• Application of Factor Theorem

10th Grade Math

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