DIVIDE POLYNOMIAL USING LONG DIVISION

Divide the following polynomials using long division.

Problem 1 :

(x² + x - 17) ÷ (x - 4)

Solution :

Step 1 :

In the first step, we are going to divide the first term of the dividend by the first term of the divisor.

After changing the signs, +x² and -x² will get canceled. By simplifying, we get 5x - 17.

Step 2 :

In the second step again we are going to divide the first term that is 5x by the first term of divisor that is x.

Quotient = x + 5

Remainder = 3

Problem 2 :

(3x² - 14x – 5) ÷ (x - 5)

Solution :

Quotient = 3x + 1

Remainder = 0

So, the given polynomial is divisible by (x - 5).

Problem 3 :

(x³ + x² + x + 2) ÷ (x² - 1)

Solution :

Quotient = x + 1

Remainder = 2x + 3

Problem 4 :

(7x³ + x² + x) ÷ (x² + 1)

Solution :

Quotient = 7x + 1

Remainder = - 6x - 1

Problem 5 :

(5x⁴ – 2x³ - 7x² - 39) ÷ (x² + 2x - 4)

Solution :

Quotient = 5x² - 12x + 37

Remainder = - 122x + 109

Problem 6 :

(4x⁴ + 5x - 4) ÷ (x² - 3x - 2)

Solution :

Quotient = 4x² + 12x + 44

Remainder = 161x + 84

Problem 7 :

The expression (6x - 5)/(x + 2) is equivalent to which of the follwoing ?

a)  6 - [17/(x + 2)]          b)  6 + [7/(x + 2)]

Solution :

By dividing the polynomial 6x - 5 by x + 2, we get

Quotient = 6 and remainder = -17

Writiting the quotient, remainder and divisor in the mixed form, we get

= 6 - [17/(x + 2)]

So, option a is correct.

Problem 8 :

When 3x² + 4 is dividied by x - 1, the result is A + [7/(x - 1)]. What is A in terms of x ?

a)  3x - 4      b)  3x - 3     c)  3x + 3     d)  3x + 4

Solution :

Quotient = 3x + 3 and remainder = 7

Writiting the quotient, remainder and divisor in the mixed form, we get

= (3x + 3) + [7/(x - 1)]

Comparing  A + [7/(x - 1)] and (3x + 3) + [7/(x - 1)], the value of A is 3x + 3. So, option c is correct.

Problem 9 :

If the expression (5x² - 4x + 1)/(x - 2) is written in the form 5x + 6 + [B/(x - 2)], where B is constant what is the value of B ?

a)  3      b)  13     c)  -4     d)  -13

Solution :

Quotient = 5x + 6 and remainder = 13

Writiting the quotient, remainder and divisor in the mixed form, we get

= (5x + 6) + [13/(x - 2)]

Comparing  5x + 6 + [B/(x - 2)] and (5x + 6) + [13/(x - 2)], the value of B is 13. So, option b is correct.

Problem 10 :

If the expression (2x² - 5x + 1)/(x - 3) is written in the equivalent form 2x + 1 + [R/(x - 3)], what is the value of R

a)  4      b)  2     c)  -4     d)  -3

Solution :

Quotient = 2x + 1 and remainder = 4

Writiting the quotient, remainder and divisor in the mixed form, we get

= (2x + 1) + [4/(x - 3)]

Comparing 2x + 1 + [R/(x - 3)] and (2x + 1) + [4/(x - 3)], the value of R is 4. So, option a is correct.

Problem 11 :

f(x) = 3x³ - kx² + 5x + 2

In the polynomial f(x) defined above, k is constant. If f(x) is divisible by x - 2. What is the value of k ?

a)  12    b)  9    c)  6      d)  3

Solution :

The given polynomial f(x) is divisible by x - 2. Since it is divisible by the linear x - 2, then x - 2 = 0

x = 2

the remainder will be 0.

f(2) = 3(2)³ - k(2)² + 5(2) + 2

0 = 3(8) - k(4) + 10 + 2

0 = 24 - 4k + 10 + 2

0 = 36 - 4k

4k = 36

k = 36/4

k = 9

Problem 12 :

The table above gives the value of polynomial p(x) for some values of x. Which of the following must be a factor of p(x) ?

a) x + 1      b)  x - 1     c)  x - 4      d)  x - 5

Solution :

p(-1) = 0, x + 1 must be a factor of p(x). So, option a is correct.

Problem 13 :

For the polynomial p(x), p(2) = 0, which of the following must be true about p(x) ?

a)  2x is a factor of p(x)      b)  2x - 2 is a factor of p(x)

c)  x - 2 is a factor of p(x)        d)  x + 2 is a factor of p(x)

Solution :

While applying x = 2, we get the remainder as 0.

x - 2 is a factor of p(x). Option c is correct.