Let-f-x-ax-1-5-1-bx-4-a-b-N-if-times-of-x-equal-62-so-what-are-possible-values-of-the-sum-a-b-

Question Number 178347 by Acem last updated on 15/Oct/22

L e t f (x) = {(a x + 1)}^{5} . {(1 + b x)}^{4}; a, b \in N

i f t i m e s o f x e q u a l 62 s o w h a t a r e p o s s i b l e v a l u e s

o f t h e s u m a, b ?

Answered by Acem last updated on 15/Oct/22

{(1 + a x)}^{5} = 1 + 5 a x + 10 a^{2} x^{2} + \dots

{(1 + b x)}^{4} = 1 + 4 b x + 6 b^{2} x^{2} + \dots

W e t a k e t i m e s x f r o m m u l t i p l i c a t i o n

(5 a + 4 b) x i . e . 5 a + 4 b = 62 \dots (1)

W e n e e d t o (a + b) s o a n d f r o m (1)

4 (a + b) = - a + 62

S i n c e a \in N \Rightarrow 4 (a + b) < 62 \Leftrightarrow a + b < 15.5

S i n c e a + b \in N \Rightarrow a + b ⩽ 15 \dots (2)

W e r e p e a t f o r b f r o m (1)

5 (a + b) = b + 62, b \in N \Rightarrow 5 (a + b) > 62 \Rightarrow a + b > 12.4

S i n c e a + b \in N \Rightarrow a + b ⩾ 13 \dots (3)

(2) & (3)

W e f i n d t h a t t h e p o s s i b l e v a l u e s o f a + b a r e :

a + b = 13, a + b = 14, a + b = 15

Commented by Tawa11 last updated on 15/Oct/22

Great sir

Commented by Acem last updated on 15/Oct/22

T h a n k s S i r

Answered by mr W last updated on 16/Oct/22

c o e f . o f x t e r m i s

5 \times 1^{4} \times a \times 1^{4} + 4 \times 1^{3} \times b \times 1^{5} = 5 a + 4 b

5 a + 4 b = 62

\Rightarrow a = 4 k + 10 ⩾ 1 \Rightarrow k ⩾ - \frac{9}{4} \Rightarrow k ⩾ - 2

\Rightarrow b = - 5 k + 3 ⩾ 1 \Rightarrow k ⩽ \frac{2}{5} \Rightarrow k ⩽ 0

a + b = 4 k + 10 - 5 k + 3 = - k + 13

{(a + b)}_{m i n} = 13 a t k_{m a x} = 0

{(a + b)}_{m a x} = 15 a t k_{m i n} = - 2

\Rightarrow a + b = 13 o r 14 o r 15

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