Question

(x+4) is a factor of x⁴ + 4x³-ax²-bx+ 24. Also, a + b = 29. Find the value of b​

Answer 1

Given :

${x}^{4} + 4 {x}^{3} - a {x}^{2} - bx + 24 \: \: ... \: \: (i)\\ \\ a + b = 29 \: \: ... \: \: (ii)$

Solution :

$x + 4 = 0 \\ \\ x = - 4.$

now substituting 'x' in equation (i)

${x}^{4} + 4 {x}^{3} - a {x}^{2} - bx + 24 \\ \\ { - 4}^{4} + 4 \times { - 4}^{3} - a \times { - 4}^{2} - b \times ( - 4) + 24 \\ \\ 256 + ( - 256) - 16a - ( - 4b) + 24 \\ \\ 256 - 256 - 16a + 4b + 24 \\ \\ - 16a + 4b + 24 \\ \\ - 16a + 4b = - 24 \: \:... \: \: (iii)$

equation (iii) divided by -4 , we get ,

4a - b = 6 ... (iv)

now , for finding b we have to solve equations (ii) and (iv)

$(a + b) + (4a - b) = 29 + 6 \\ \\ a+ b+ 4a - b = 35 \\ \\ 5a = 35 \\ \\ a = \frac{35}{5} \\ \\ a = 7.$

now , putting 'a' in equation (ii)

$a + b = 29 \\ \\ 7 + b = 29 \\ \\ b = 29 - 7 \\ \\ b = 22.$

• Therefore the value b is 22 .

Answer 2

Answer:

The value of b is 22.

Step-by-step explanation:

Given polynomial: x⁴ + 4x³- ax²- bx + 24

Also, a + b = 29

To find: Value of b

Solution:

(x + 4) is a factor of the given polynomial.

So, x = –4

p(x) = x⁴ + 4x³- ax²- bx + 24

p(-4) = (–4)⁴ + 4(–4)³- a(–4)²- b(–4) + 24

= 256 - 256 - 16a + 4b + 24

= -16a + 4b + 24 --- (I)

Also, a + b = 29

a = 29 - b --- (II)

Substitute eq. (II) in eq. (I),

– 16(29 - b) + 4b + 24 = 0

– 464 + 16b + 4b + 24 = 0

20b – 440 = 0

20b = 440

b = 440/20

b = 22

Therefore, the value of b is 22.