Question

[Maths]

What number must be subtracted from $\bold{2 {x}^{2} - 5x}$so that the resulting polynomial
leaves the remainder 2 when divided by 2x +1?

Correct Answer:- 1

• Kindly Match Your Answer with the above given Answer Before Posting The Solution.
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Answer 1

Answer in the attachment if you can understand the handwriting...

Answer 2

Topic :-

Remainder Theorem

Given :-

When a number is subtracted from 2x² - 5x then resulting polynomial leaves remainder 2 when divided by 2x + 1.

To Find :-

Number that should be subtracted.

Solution :-

Let us assume number that should be subtracted be k.

p(x) = 2x² - 5x - k

Method 1

Remainder Theorem

Let p(x) be a polynomial of degree ≥ 1 and 'a' is any real number. If p(x) is divided by (x - a), then remainder is p(a).

p(x) is divided by (2x + 1) then 'a' equals to

2x + 1 = 0

2x = -1

x = -1/2 will act as 'a'.

So, to obtain remainder of p(x), put x = a.

p(x) = 2x² - 5x - k

p(-1/2) = 2(-1/2)² - 5(-1/2) - k

p(-1/2) = 2(1/4) + (5/2) - k

p(-1/2) = (1/2) + (5/2) - k

p(-1/2) = ((1+5)/2) - k

p(-1/2) = 6/2 - k

p(-1/2) = 3 - k

p(-1/2) = 2 (given)

2 = 3 - k

k = 3 - 2

k = 1

Hence, number that should be subtracted is 1.

Method 2

Long Division Method

Refer to the attachment.

Method 3

Manipulation

We will manipulate the given data such that we get value of 'k'.

Dividend = Divisor × Quotient + Remainder

p(x) = (2x + 1)Q + 2

2x² - 5x - k = (2x + 1)Q + 2

2x² - 5x - k - 2 = (2x +1)Q

Now, we will split -5x in such a way such that we get (2x + 1) as one of the factor.

2x² - 6x + x - k - 2 = (2x + 1)Q

2x(x - 3) + x - k - 2 = (2x + 1)Q

Now, we need to add and subtract a number which can help in making (x - 3) as a factor.

2x(x - 3) + x - 2 - 1 + 1 - k = (2x + 1)Q

2x(x - 3) + 1(x - 3) + 1 - k = (2x + 1)Q

(x - 3)(2x + 1) + 1 - k = (2x + 1)Q

Now, LHS is whole divisible by (2x + 1) which means

1 - k = 0

k = 1

Answer :-

Hence, number that should be subtracted is 1.