Question

if (X + 2) is a factor of X⁵ - 4a²x + 2 X +2a+3 find a.

Answer 1

Answer:

Let p(x) = x5 − 4a2x3 + 2x + 2a + 3

Given (x + 2a) is a factor of p(x)

Hence p(−2a) = 0 by remainder theorem

Put x = −2a in p(x)

p(−2a) = (−2a)5 − 4a2(−2a)3 + 2(−2a) + 2a + 3

      0  = −32a5 − 4a2(−8a3) − 4a + 2a + 3

⇒ −32a5 + 32a5 − 2a + 3 = 0

⇒ − 2a + 3 = 0

⇒ 2a  = 3

∴ a = (3/2)

Step-by-step explanation:

welcome

Answer 2

Step by step Explanation:

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

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+2=0 and find the value of x

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

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

${x}^{5} - 4 {a}^{2} {x} + 2x + 2a + 3 = 0 \\ \\ { (- 2)}^{5} - 4 {a}^{2} {( - 2)} + 2( - 2)+ 2a + 3 = 0 \\ \\ - 32 +8 {a}^{2} - 4+ 2a + 3 = 0$

Simplify the expression

$8 {a}^{2} +2 {a}- 33 = 0...eq3$

Step 3: Solve the quadratic equation in a

Apply Quadratic formula

$a_{1,2}=\frac{-b±\sqrt{b^2-4ac}}{2a}\\ \\ a_{1,2}=\frac{-2±2\sqrt{265}}{16} \\ \\ a_{1,2}=\frac{-1±\sqrt{265}}{8} \\ \\ a_1 =1.90 \\ \\a_2=-2.16$

Final answer:

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

$\boxed{\bold{\green{a = 1.90}\:\quad or\:\quad\red{a=-2.16} }}$

Hope it helps you.

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