Find the quotient and remainder by dividing3x 2 +13x 2 +9x+6 by 3x-1 determine t

Question

Find the quotient and remainder by dividing3x2+13x2+9x+6 by 3x-1
determine the polynomial f(x), that when divided by 3x+4yields a quoitient polynomial q(x)= 2x2+5x-3 withremainder 3
Find a quadratic polynomial for which 4-2i and 4+2i arecomplex roots
Find a polynomial f(x) of degree 3 such that 1-3i, and 4arezeros


determine the polynomial f(x), that when divided by 3x+4yields a quoitient polynomial q(x)= 2x2+5x-3 withremainder 3
Find a quadratic polynomial for which 4-2i and 4+2i arecomplex roots
Find a polynomial f(x) of degree 3 such that 1-3i, and 4arezeros

Explanation / Answer

This must be a type-o determine the polynomial f(x), that when divided by 3x+4yields a quoitient polynomial q(x)= 2x2+5x-3 withremainder 3 Multiply (3x +4) by (2x2+5x-3) then add 3 (3x +4)   (2x2+5x-3) =6x3 +15x2 -9x +8x2 +20x -12 =6x3 +23x2 +11x -12 +3 6x3+23x2 +11x -9 Find a quadratic polynomial for which 4-2i and 4+2i arecomplex roots If those are roots then the factors are (x-4+2i)(x-4-2i)=0 multiply factors x2 -4x -2ix      -4x +    +8i  +16            +2ix -8i -4i2             since i2 = -1 then  -4i2 = +4 x2 -8x +20 x2 -8x +20 F(x) = x2 -8x +20 Find a polynomial f(x) of degree 3 such that 1-3i, and 4arezeros if 1- 3i is a zero then its conjugate ( 1+3i) is a zeroas well so the factors are; (x-1+3i)(x-1 -3i)(x-4) multiply to get thefunction (x-1+3i)(x-1 -3i)= x2   -x   - 3ix       -x            +1 +3i             + 3ix        -3i  -9i2   since i2 = -1then  -9i2 = +9 (x2 -2x +10) Now we multiply by (x-4) (x-4)(x2 -2x +10) = x3-2x2 + 10x -4x2 +8x -40 f(x) =  x3-6x2 + 18x -40

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