Question

If p = ( 2 - a ) , then a³ + 6ap + p³ - 8 is ?

Answer 1

Given:

• p = (2 - a)

To find:

• We need to find the value of a³ + 6ap + p³ - 8.

Solution:

In this problem, we are given that p = (2 - a). From this, we can say that:

$\leadsto$ p = 2 - a

$\leadsto$ p + a = 2

We need to find the value of a³ + 6ap + p³ - 8.

$\leadsto$ a³ + 6ap + p³ - 8

$\leadsto$ a³ + p³ + 6ap - 8

Now, we are going to use an identity. We know that:

$\longrightarrow$ a³+p³ = (a+p)(a²-ap+p²)

Let us apply the above identity:

$\leadsto$ (a+p)(a²-ap+p²) + 6ap - 8

$\leadsto$ 2(a²-ap+p²) + 6ap - 8 [a+p = 2]

$\leadsto$ 2a²-2ap+2p²+6ap - 8

$\leadsto$ 2a²+2p²+4ap - 8

$\leadsto$ 2(a²+p²+2ap) - 8

We know that,

$\longrightarrow$ (a+p)² = a²+p²+2ap

On applying the above identity, we get:

$\leadsto$ 2(a+p)² - 8

$\leadsto$ 2(2)² - 8 [a+p = 2]

$\leadsto$ 2×4 - 8

$\leadsto$ 8 - 8

$\leadsto$ 0

Conclusion:

Hence, the value of a³ + 6ap + p³ - 8 is 0.

Answer 2

Step-by-step explanation:

Given:

p = (2 - a)

To find:

We need to find the value of a³ + 6ap + p³ - 8.

Solution:

In this problem, we are given that p = (2 - a). From this, we can say that:

⇝ p = 2 - a

⇝ p + a = 2

We need to find the value of a³ + 6ap + p³ - 8.

⇝ a³ + 6ap + p³ - 8

⇝ a³ + p³ + 6ap - 8

Now, we are going to use an identity. We know that:

\longrightarrow⟶ a³+p³ = (a+p)(a²-ap+p²)

Let us apply the above identity:

⇝ (a+p)(a²-ap+p²) + 6ap - 8

⇝ 2(a²-ap+p²) + 6ap - 8 [a+p = 2]

⇝ 2a²-2ap+2p²+6ap - 8

⇝ 2a²+2p²+4ap - 8

⇝ 2(a²+p²+2ap) - 8

We know that,

\longrightarrow⟶ (a+p)² = a²+p²+2ap

On applying the above identity, we get:

\leadsto⇝ 2(a+p)² - 8

\leadsto⇝ 2(2)² - 8 [a+p = 2]

\leadsto⇝ 2×4 - 8

\leadsto⇝ 8 - 8

\leadsto⇝ 0

Conclusion:

Hence, the value of a³ + 6ap + p³ - 8 is 0.