Answer:
Step-by-step explanation:
To find the value of a, we need to use the fact that (x + 2) is a factor of the given polynomial.
When (x + 2) is a factor of a polynomial, it means that if we substitute (-2) for x in the polynomial, the result should be zero.
Let's substitute (-2) for x in the given polynomial:
(-2)^5 - 4a^2(-2)^3 + 2(-2) + 2a + 3 = 0
-32 + 32a^2 - 4 - 2a + 2a + 3 = 0
Combining like terms, we get:
32a^2 - 33 = 0
Simplifying further, we have:
32a^2 = 33
Dividing both sides by 32, we get:
a^2 = 33/32
Taking the square root of both sides, we get:
a = ±√(33/32)
Therefore, the value of a is ±√(33/32).
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Answers & Comments
Given :
[tex]x + 2 = 0 \\ \\ x = - 2[/tex]
To Find :
[tex]The \: Factor \: of \: \: {x}^{5} - 4 {a}^{2} {x}^{3} \\ + 2x + 2a + 3[/tex]
Solution Explanation :
[tex]p(x) = {x}^{5} - 4 {a}^{2} {x}^{3} + 2x \\ + 2a + 3 \\ \\ p( - 2) = {( - 2)}^{5} - 4 {a}^{2} {( - 2)}^{3} \\ + 2( - 2) + 2a + 3 \\ \\ = ( - 32) - 4 {a}^{2} ( - 8) - 4 \\ + 2a + 3 \\ \\ = - 32 + 32{a}^{2} + 2a - 1 \\ \\ = - {a}^{2} + 2a - 1 \\ \\ = - {a}^{2} + (1 + 1)a - 1 \\ \\ = - {a}^{2} + a + a - 1 \\ \\ = - a(a - 1) +1 (a - 1) \\ \\ = ( - a + 1) (a - 1) \\ \\ - a + 1 = 0 \: \: \: \: \: \: \: \: a - 1 = 0 \\ \\ - a = - 1 \: \: \: \: \: \: a = 1 \\ \\ a = 1 \: \: \: \: \: \: \: a = 1[/tex]
[tex] \\ [/tex]
[tex]The \: value \: of \: a \: is \: 1[/tex]
Answer:
Step-by-step explanation:
To find the value of a, we need to use the fact that (x + 2) is a factor of the given polynomial.
When (x + 2) is a factor of a polynomial, it means that if we substitute (-2) for x in the polynomial, the result should be zero.
Let's substitute (-2) for x in the given polynomial:
(-2)^5 - 4a^2(-2)^3 + 2(-2) + 2a + 3 = 0
-32 + 32a^2 - 4 - 2a + 2a + 3 = 0
Combining like terms, we get:
32a^2 - 33 = 0
Simplifying further, we have:
32a^2 = 33
Dividing both sides by 32, we get:
a^2 = 33/32
Taking the square root of both sides, we get:
a = ±√(33/32)
Therefore, the value of a is ±√(33/32).