Question

a polynomial f(x) leaves reminder 10 and 14, when divided by (x-3) and (x-5) find the remainder when f(x) is divided by (x-3)(x-5)

Answer 1

solve: (x-3) (x-5)

( 10-3) ( 14-5)

7x 9

63(ans)

I thik it helps you

Answer 2

Answer:

-2x+16

Step-by-step explanation:

Given, p(x) when divided by x−3 and x−5, leaves remainder 10 and 6, respectively.

From remainder theorem of polynomial, p(3)=10 and P(5)=6.

If the polynomial is divided by (x−3)(x−5), then the remainder must be of the form ax+b (Since, degree of remainder is less than that of the divisor).

⇒p(x)=q(x)(x−3)(x−5)+(ax+b), where q(x) is some polynomial (quotient obtained).  

Substituting for x=3 and x=5:

p(3)=q(x)(3−3)(3−5)+a(3)+b=0+3a+b=3a+b⟹10=3a+b−−(i)

p(5)=q(x)(5−3)(5−5)+a(5)+b=0+5a+b=5a+b⟹6=5a+b−−(ii).

Solving for a and b, we get,

(i)−(ii)⟹10−6=(3−5)a⟹4=−2a⟹a=−2−−(a).

(a) in (ii)⟹6=5(−2)+b⟹6=−10+b⟹b=6+10⟹b=16.

∴a=−2 and b=16.

⇒ Remainder = −2x+16

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