Question

The polynomials P(t) = 4t3 - st2 + 7 and Q(t) = t2 + st + 8 leave the same remainder when divided by (t - 1). Find the value of s.

Answer 1

Step-by-step explanation:

Remainder theorem: If a polynomial P(x) is divided by (x-c), then the remainder is equal to P(c).

The given polynomial are

p(t)=4t^3-st^2+7p(t)=4t

3

−st

2

+7

q(t)=t^2+st+8q(t)=t

2

+st+8

Using remainder theorem the remainder of \frac{p(t)}{t-1}

t−1

p(t)

is p(1) and the remainder of \frac{q(t)}{t-1}

t−1

q(t)

is q(1).

Substitute t=1 in the given functions.

p(1)=4(1)^3-s(1)^2+7\Rightarrow 4-s+7=11-sp(1)=4(1)

3

−s(1)

2

+7⇒4−s+7=11−s

q(1)=(1)^2+s(1)+8=1+s+8=9+sq(1)=(1)

2

+s(1)+8=1+s+8=9+s

It is given that if p(t) and q(t) divided by (t-1), then the remainder is same.

p(1)=q(1)p(1)=q(1)

Substitute these values.

11-s=9+s11−s=9+s

Add s on both sides.

11=9+s+s11=9+s+s

11=9+2s11=9+2s

Subtract 9 from both sides.

11-9=2s11−9=2s

2=2s2=2s

Divide both sides by 2.

1=s1=s