Polynomials Test - 5

Area of the rectangle = Length × breadth
= (4x + 3) × (2x – 5)

= 8x² – 20x + 6x – 15
= (8x² – 14x – 15) m²

Let f(x) = x⁴ + 2x³ - 2x² + x - 1 is divided by x² + 2x - 3.

x⁴ + 2x³ - 2x² + x -1 = {(x² + 2x -3) (x² + 1)} - x + 2

i.e. we get remainder a= -x + 2.

Hence, x - 2 must be added to x⁴ + 2x³ - 2x² + x - 1, so that the result is exactly divisible by x² + 2x - 3.

i.e x - 2 should be added.

Amount of money Rahul has = Rs. (x³ + 5x² + px + q)

Now, he can buy exactly(x - 2) balls or (x + 2) gloves.

Therefore,(x - 2) and (x + 2) are the factors of (x³ + 5x² + px + q).

Therefore, (2)³ + 5(2)² + 2p + q = 0

8 + 20 + 2p + q = 0

2p + q = -28 --- (1)

And (-2)³ + 5(-2)² - 2p + q = 0

-8 + 20 - 2p + q = 0

-2p + q = -12 --- (2)

On adding (1) and (2), we get

2q = -40

q = -20

From Remainder Theorem,
At x = 2, the remainder is 8 - 4p - 8 + 12 = 12 - 4p.

At x = 3, the remainder is 27 - 9p - 12 + 12 = 27 - 9p.
As the remainders are same;

12 - 4p = 27 - 9p
5p = 15
p = 3

Statement 1:
(x + 6) and (x – 4) are factors of the expression 2x³ + 2ax² + 2bx – 24.

f(x) = 2x³ + 2ax² + 2bx – 24
Put x = 4 in the equation.

f(4) = 2(4)³ + 2a(4)² + 2b(4) – 24 = 0
128 + 32a + 8b – 24 = 0

32a + 8b + 104 = 0
4a + b + 13 = 0 ... (I)

Now, put x = -6 in the given equation.
2(-6)³ + 2a(-6)² + 2b(-6) – 24 = 0

-432 + 72a - 12b - 24 = 0
-36 + 6a - b – 2 = 0
6a – b – 38 = 0 ... (II)

Adding equations (I) and (II),
4a + b +13 + 6a – b – 38 = 0

10a – 25 = 0
2a – 5 = 0, a = 5/2

Put the value of 'a' in equation (I).
4a + b + 13 = 0

4(2.5) + b +13 = 0
10 + b + 13 = 0

b = -23
So, statement 1 is true.

Statement 2:
2x³ – 134x + 252 ... (i)

As we know, (2x – 14) is a factor and we need to find two more factors for the
expression, we will break 134x in such a way that it is easy to factorise.
134x = 98x + 36x

Also, adding and subtracting 14x² in the expression (i),
2x³ + 14x² – 14x² – 98x - 36x + 252
= (2x³ – 14x²) + (14x² – 98x) – (36x – 252)

= x²(2x – 14) + 7x(2x – 14) – 18(2x – 14)
= (2x – 14)(x² + 7x – 18)

= (2x – 14)(x² + 9x – 2x – 18)
= (2x – 14)[x(x + 9) - 2(x + 9)]
= (2x – 14)(x – 2)(x + 9)

So, both the statements are true.