Question

If x + 2a is a factory of x5 - 4a2 + 2x + 2a +3, find a​

Answer 1

Step-by-step explanation:

Let p(x) = a5 -4a2x3 +2x + 2a +3

Since, x + 2a is a factor of p(x), then put p(-2a) = 0 (-2a)5 – 4a2 (-2a)3 + 2(-2a) + 2a + 3 = 0 ⇒ -32a5 + 32a5 -4a + 2a+ 3 = 0

⇒ -2a + 3 = 0

2a =3

a = 3/2.

Hence, the value of a is 3/2.

Answer 2

Step-by-step explanation:

Given: If x+ 2a is a factor of $x^5 -4a^2x^3 + 2x+ 2a +3$,

To find: Find the value of a.

Solution:

Tip: Apply factor theorem.

It states that,if (x-a) is a factor of polynomial f(x), then f(a)=0

Step 1: Put x+2a=0 and find the value of x

$x = - 2a \: ...eq1$

Step 2: Put the value of eq1 in polynomial and find a

${x}^{5} - 4 {a}^{2} {x}^{3} + 2x + 2a + 3 = 0 \\ \\ { (- 2a)}^{5} - 4 {a}^{2} {( - 2a)}^{3} + 2( - 2a)+ 2a + 3 = 0 \\ \\ - 32 {a}^{5} + 32 {a}^{2} {a}^{3} - 4a + 2a + 3 = 0$

because

$\boxed{\bold{ {x}^{n} . {x}^{m} = {x}^{n + m} }}$

Apply the formula

$- 32 {a}^{5} +32 {a}^{5}- 2a + 3 = 0$

Cancel same terms having opposite signs

$\cancel{- 32 {a}^{5}} +\cancel{ 32 {a}^{5}} - 2a + 3 = 0$

$- 2a + 3 = 0 \\ \\ - 2a = - 3 \\ \\ 2a = 3 \\ \\ a = \frac{3}{2}$

Final answer:

If x+ 2a is a factor of $x^5 -4a^2x^3 + 2x+ 2a +3$,then

$\boxed{\bold{\purple{a = \frac{3}{2}} }}$

Hope it helps you.

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