PP Section 12.3

# Example polynomial the of zeros are 1 and 2 3 fact

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Example . polynomial the of zeros are 1 and 2, - 3, - fact that the Use . 12 20 9 ) ( Factor 2 3 4 5 - + - - + = x x x x x x f We can use synthetic division with the given roots to start factoring the polynomial. -3 1 1 -9 -1 20 -12 1 -2 -3 8 -4 0 -3 6 9 -24 12 Got: ( x + 3 )( x 4 – 2x 3 – 3x 2 + 8x – 4 )

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-2 needs to be a root of the quotient found, so we apply synthetic division to that polynomial. -2 1 -2 -3 8 -4 1 -4 5 -2 0 -2 8 -10 4 Got: ( x + 3 )( x + 2)( x 3 – 4x 2 + 5x – 2 ) 1 1 -4 5 -2 1 -3 2 0 1 -3 2 Got: ( x + 3 )( x + 2)( x – 1 )( x 2 - 3x + 2 )
To complete the factorization, all we need to do is factor the quadratic polynomial found. Got: ( x + 3 )( x + 2)( x – 1 )( x 2 - 3x + 2 ) Therefore, f(x) = (x + 3)(x + 2)(x – 1)(x – 2)(x – 1) OR: f(x) = (x + 3)(x + 2)(x – 1) 2 (x – 2) Looking at the factorization of f(x) , we see that –3, -2, and 2 are roots of multiplicity one, and 1 is a root of multiplicity 2.

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Example Find a polynomial f(x) such that x – 3 is a factor of the polynomial and –5 is a root of the polynomial of multiplicity 2. f(x) = (x – 3)(x + 5) 2 Is this answer unique? No. The degree of the polynomial was not specified. Also, it could have other roots since the problem did not indicate that the given ones were the only roots.
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