Synthetic Division

Question

1. If x-5 and x+4 are factors of x^4-4x^3-15x^2+58x-40, find the remaining linear factor
2. One of the roots of f(x)= 2x^3-17x^2+12x+63 is 7. Find the remaining roots
3. Two of the roots of f(x)= 8x^4-14x^3-72x^2-10x+24 are -2 and 4. Find the remaining roots

Answer 1

1) then 5 and -4 are roots

5 | 1 -4 -15 58 -40

5 5 -50 40

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1 1 -10 8 0

-4 | 1 1 -10 8

-4 12 -8

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1 -3 2 0

the quotient is x^2 - 3x + 2

= ( x - 2 )(x - 1 )

THe other two solutions are x=2 and x = 1

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2)

7 | 2 -17 12 63

14 -21 -63

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2 -3 -9 0

the quotient is 2x^2 - 3x - 9

( 2x + 3 )( x - 3 )

the solutions are x = -3/2 and x = 3

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3)

-2 and 4 ARE NOT ROOTS !!! THere is a typo

p = {factors of 24} = {+-1, +-2, +-3, +-4, +-6, +-8 , +-12, +-24}

q = {factors of 8} = {+-1, +-2, +-4, +-8}

the possible rational solutons are p/q =

{+-1, +-1/2 , +-1/4, +-1/8, +-2, +-3, +-3/2, +-3/4, +-3/8,

+-4, +-6, +-8, +-12, -24}

NONE of them work!!! So there are no rational solutions.

The solutions must then be found numerically, by inspecting the graph

Per Desmos, the solutions are APPROXIMATELY

-2.026, -0.741, 0.497, 4.02