Find the values of a and b so that the polynomial (x4+ax3−7x2+8x+b) is exactly divisible by (x+2) ad well as (x+3).

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Filo · Accepted Answer

We are given that polynomial (x4+ax3−7x2+8x+b) is exactly divisible by (x+2) as well as (x+3).The two divisors in the question are (x + 2) and (x + 3). As it is given that the given polynomial is divisible by both these divisors, that means;x+2=0   and   x+3=0x=−2  and  x=−3 will make the remainder zero when these values of x are substituted in the given polynomial.f(x)=x4+ax3−7x2+8x+bSo, f(−2) and f(−3) will be equal to zero.f(−2)=(−2)4+a(−2)3−7(−2)2+8(−2)+b=0f(−3)=(−3)4+a(−3)3−7(−3)2+8(−3)+b=0f(−2)=16−8a−28−16+b=0 ---------------- [Equation 1]f(−3)=81−27a−63−24+b=0−27a+b−6=0  ---------------- [Equation 2]Now using the elimination method to find the values of a and b;−8a+b−28=0−27a+b−6=0+        -     +       -       \overline{\ \ 19a - \ \ 22 \ \ =\ \  0}     a=1922​Putting the value of a in equation 1 we get;−8a+b−28=0−8(1922​)+b−28=0                            b = 28+19176​                            b = 19708​Hence, the value of a=1922​ and the value of   b = 19708​

Question Text	Find the values of a and b so that the polynomial $(x^{4} + a x^{3} - 7 x^{2} + 8 x + b)$ is exactly divisible by $(x + 2)$ ad well as $(x + 3)$ .
Answer Type	Text solution:1
Upvotes	150

Find the values of a and b so that the polynomial $(x^{4} + a x^{3} - 7 x^{2} + 8 x + b)$ is exactly divisible by $(x + 2)$ ad well as $(x + 3)$ .

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