Question

In each of the following two polynomials, find the value of a, if x+a is a factor:
(i) x³+ax²-2x+a+4
(ii) x⁴-a²x²+3x-a

Answer 1

(i) The value of a is $\frac{-4}{3}$

(ii) The value of a is 0

Step-by-step explanation:

Given polynomials are (i) $x^3+ax^2-2x+a+4$

(ii) $x^4-a^2x^2+3x-a$

Also given that x+a is a factor for the given two polynomials.

To find the value of a in the polynomial :

(i) $x^3+ax^2-2x+a+4$

By using the Synthetic Division we can solve this cubic expression.

$x^3+ax^2-2x+a+4=0$ since x+a is a factor so it satisfies the given equation

 -a_|  1     a     -2      a+4

        0    -a      0       2a

      _________________

       1      0     -2       a+4+2a=0

• a+4+2a=0

• 3a+4=0
• $3a=-4$
• $a=\frac{-4}{3}$

Therefore the value of a is $\frac{-4}{3}$

(ii) $x^4-a^2x^2+3x-a$

By using the Synthetic Division we can solve this  expression.

$x^4+0x^3-a^2x^2+3x-a=0$ since x+a is a factor so it satisfies the given equation

 -a_|  1        0    $-a^2$    3      -a

         0      -a      $a^2$     0     -3a

       __________________________

         1      -a        0     3     -4a=0

• -4a=0

• $a=\frac{0}{-4}$
• a=0

Therefore the value of a is 0

Answer 2

Answer:

1)

Step-by-step explanation:

Let p(x) = x³ + ax² - 2x + a + 4 ... (i) X Since, (x + a) is a factor of p(x), so p(-a)

Let p(x) = x³ + ax² - 2x + a + 4 ... (i) X Since, (x + a) is a factor of p(x), so p(-a)= 0

Let p(x) = x³ + ax² - 2x + a + 4 ... (i) X Since, (x + a) is a factor of p(x), so p(-a)= 0Put x = -a in equation (i), we get p(-a) = (-a)³ + a(-a)²-2(-a) + a + 4 = -a³ + a(a²) + 2a + a + 4 = -a³ + a³ + 3a + 4 = 3a + 4

Let p(x) = x³ + ax² - 2x + a + 4 ... (i) X Since, (x + a) is a factor of p(x), so p(-a)= 0Put x = -a in equation (i), we get p(-a) = (-a)³ + a(-a)²-2(-a) + a + 4 = -a³ + a(a²) + 2a + a + 4 = -a³ + a³ + 3a + 4 = 3a + 4But p(-a) = 0 → 3a + 4 = 0

Let p(x) = x³ + ax² - 2x + a + 4 ... (i) X Since, (x + a) is a factor of p(x), so p(-a)= 0Put x = -a in equation (i), we get p(-a) = (-a)³ + a(-a)²-2(-a) + a + 4 = -a³ + a(a²) + 2a + a + 4 = -a³ + a³ + 3a + 4 = 3a + 4But p(-a) = 0 → 3a + 4 = 0→ 3a = -4

→ a = -4/3

2)

Let f(x) = x4-a²x²+3x-a

Let f(x) = x4-a²x²+3x-a(x+a) is a factor of f (x)

Let f(x) = x4-a²x²+3x-a(x+a) is a factor of f (x)f (-a) = 0

Let f(x) = x4-a²x²+3x-a(x+a) is a factor of f (x)f (-a) = 0(-a) 4-a²(-a)²+3(-a)-a = 0

Let f(x) = x4-a²x²+3x-a(x+a) is a factor of f (x)f (-a) = 0(-a) 4-a²(-a)²+3(-a)-a = 0a4-a4-3a-a = 0

Let f(x) = x4-a²x²+3x-a(x+a) is a factor of f (x)f (-a) = 0(-a) 4-a²(-a)²+3(-a)-a = 0a4-a4-3a-a = 0-4a = 0

Let f(x) = x4-a²x²+3x-a(x+a) is a factor of f (x)f (-a) = 0(-a) 4-a²(-a)²+3(-a)-a = 0a4-a4-3a-a = 0-4a = 0a = 0

Hope it is helpful