Mathematics

Question

Find the values of a and b given that the polynomial P(x) = x^3+ ax^2 +x+b, is divisible by
both x-1 and x + 3.​

1 Answer

  • Answer:

    a=4

    b=-6

    Step-by-step explanation:

    If P(x) is divisible by x-c, then P(c)=0.

    So P(1)=0 implies 1^3+a1^2+1+b=0

    and

    P(-3)=0 implies (-3)^2+a(-3)^2+-3+b=0

    So notice we have a system to solve.

    Let's simply it.

    First equation:

    1^3+a1^2+1+b=0

    1+a+1+b=0

    2+a+b=0

    a+b=-2

    Second equation:

    (-3)^3+a(-3)^2+-3+b=0

    -27+9a-3+b=0

    9a+b-30=0

    9a+b=30

    Let's put our system together:

    a+b=-2

    9a+b=30

    This is setup so if we subtract the equations b will eliminate allowing us to solve for a:

    -8a=-32

    a=4

    If a=4 and a+b=-2, then 4+b=-2 giving us b=-6.

    a=4

    b=-6

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