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Two of the zeroes of the given polynomial are . Therefore two of the factors of the given polynomial are  and     is a factor of the given polynomial. To find the other factors we divide the given polynomial with    The quotient we have obtained after performing the division is  (x+1)2 = 0 x = -1 The other two zeroes of the given polynomial are -1.
The polynomial division is carried out as follows  Given the remainder =x+a The obtained remainder after division is now equating the coefficient of x which gives the value of now equating the constants Therefore k=5 and a=-5
Given the two zeroes are   therefore the factors are We have to find the remaining two factors. To find the remaining two factors we have to divide the polynomial with the product of the above factors Now carrying out the polynomial division Now we get So the zeroes are
The roots of the above polynomial are a, a - b and a + b Sum of the roots of the given plynomial = 3 a + (a - b) + (a + b) = 3 3a = 3 a = 1 The roots are therefore 1, 1 - b and 1 + b Product of the roots of the given polynomial = -1 1 x (1 - b) x (1 + b) = - 1 1 - b2 = -1 b2 - 2 = 0 Therefore a = 1 and .
Let the roots of the polynomial be  Hence the required cubic polynomial is x3 - 2x2 - 7x + 14 = 0
p(x) = x3 - 4x2 + 5x - 2 p(2) = 23 - 4 x 22 + 5 x 2 - 2 p(2) = 8 - 16 + 10 - 2 p(-2) = 0 p(1) = 13 - 4 x 12 + 5 x 1 - 2 p(1) = 1 - 4 + 5 - 2 p(1) = 0 Therefore the numbers given alongside the polynomial are its zeroes Verification of relationship between the zeroes and the coefficients Comparing the given polynomial with ax3 + bx2 + cx + d, we have a = 1, b = -4, c = 5, d = -2 The roots...
p(x) = 2x3 + x2 -5x + 2 p(1) = 2 x 13 + 12 - 5 x 1 + 2 p(1) =2 + 1 - 5 + 2 p(1) = 0 p(-2) = 2 x (-2)3 + (-2)2 - 5 x (-2) +2 p(-2) = -16 + 4 + 10 + 2 p(-2) = 0 Therefore the numbers given alongside the polynomial are its zeroes Verification of relationship between the zeroes and the coefficients Comparing the given polynomial with ax3 + bx2 + cx + d, we have a = 2, b = 1, c = -5, d = 2 The...
example for the polynomial which satisfies the division algorithm with r(x)=0 is given below
Example for a polynomial  with deg q(x) = deg r(x) is given below
deg p(x) will be equal to the degree of q(x) if the divisor is a constant. For example
Quotient = x-2 remainder =-2x+4 Carrying out the polynomial division as follows
The polynomial division is carried out as follows The remainder is not zero, there for the first polynomial is not a factor of the second polynomial
To check whether the first polynomial is a factor of the second polynomial we have to get the remainder as zero after the division After division, the remainder is zero thus  is a factor of
After dividing we got the remainder as zero. So  is a factor of
The polynomial is divided as follows The quotient is  and the remainder is
The division is carried out as follows The quotient is  and the remainder is 8
The polynomial division is carried out as follows The quotient is x-3 and the remainder is 7x-9
The required quadratic polynomial is x2 - 4x + 1
The required quadratic polynomial is 4x2 + x + 1
The required quadratic polynomial is x2 - x + 1
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