Polynomials Test - 4

Let p(x) = 3x³ - ax² - 2bx - 7 and g(x) = x – 2
On division of p(x) by g(x), remainder = 5; this implies that p(x) - 5 is divisible by g(x).

⇒ p(2) - 5 = 0 (By Remainder Theorem)
3(2)³ - a(2)² - 2b(2) - 7 = 5

3 × 8 - 4a - 4b - 7 = 5
24 - 4(a + b) - 7 = 5

17 - 4(a + b) = 5
-4(a + b) = 5 - 17
a + b = 3

(x² – y²)(x² – 2xy + y²) = 3
Or, (x – y)(x + y)(x² – 2xy + y²) = 3

Or, (x – y)(x + y)(x – y)² = 3
x + y = 3

We have, x + y = 3 and x – y = 1
Adding two, we get 2x = 4

i.e. x = 2
and y = 1

Value of xy = 2

(a² + 9b²)(a + 3b)(a - 3b)
= (a² + 9b²)(a² - 9b²)

= (a²)² - (9b²)²
= a⁴ - 81b⁴

Let p(x) = 2x³ + 3x² - 14x - 20

When the expression p(x) is divided by (x - 4), then by Remainder Theorem, the remainder will be p(4).

So, the remainder is:
p(4) = 2(4)³ + 3(4)² - 14(4) - 20
= 128 + 48 - 56 - 20
= 176 - 76
= 100

Volume of the cuboid = Length × breadth × height
= (2p + q) × (2p - q) × (4p² + q²)

= (4p² – 2pq + 2pq – q² )(4p² + q²)
= (4p² - q²) × (4p² + q²)
= (16p⁴ - q⁴) cm³