Question

find the value of A and B so that the polynomial x cube + 10 X square + a x + b is exactly divisible by x minus 1 and X + 2​

Answer 1

Answer:

a = 7

b = -18

Step-by-step explanation:

$p(x) = x^3+10x^2+ax+b$

If (x - 1) is a factor of p(x), then p(1) = 0.

$p(1)=1^3+10(1^2)+a(1)+b=0 \\ \\ p(1)=1+10+a+b=0 \\ \\ p(1)=a+b+11=0 \ \ \ \ \ \longrightarrow \ \ \ \ \ (1)$

If (x + 2) is a factor of p(x), then p(-2) = 0.

$p(-2)=(-2)^3+10((-2)^2)+a(-2)+b=0 \\ \\ p(-2)=-8+10(4)-2a+b=0 \\ \\ p(-2)=-8+40-2a+b=0 \\ \\ p(-2)=b-2a+32=0 \ \ \ \ \ \longrightarrow \ \ \ \ \ (2)$

From (1) and (2),

$a+b+11=b-2a+32 \\ \\ a+11=32-2a \\ \\ 2a+a=32-11 \\ \\ 3a=21 \\ \\ a=\frac{21}{3} = \bold{7}$

From (1),

$a+b+11=0 \\ \\ 7+b+11=0 \\ \\ b+18=0 \\ \\ b = \bold{-18}$

Hope this may be helpful.

Thank you. Have a nice day. :-)

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