polynomials 10

• Finding zeros of a polynomial

  Set the polynomial equal to 0, factorize the polynomial, solve the equation to find the zeros or roots
• Zeros of a polynomial

  Values of X that make the polynomial equal to 0
• Roots of a polynomial

  Values of X that make the polynomial equal to 0
• Relation between zeros and coefficients of a quadratic polynomial

  Sum of the roots is -B/a, product of the roots is C/a
• If we know the zeros of a polynomial, we can write the quadratic polynomial as X^2 - (sum of roots)X + (product of roots)
• To obtain the zeros of a polynomial, set the polynomial equal to 0, factorize it, and solve the equation
• Verify the relation between the zeros and coefficients of the polynomial root 3x^2 - 8x + 4 root 3
• The sum of the roots of a quadratic polynomial is given by the formula alpha plus beta equals -B/a
• The product of the roots of a quadratic polynomial is given by the formula alpha beta equals C/a
• The roots of the polynomial are X = 2/root 3 and X = 2 root 3
• The sum of the roots is 8 root 3
• The product of the roots is 4
• Alpha plus beta

  8 root 3
• Alpha beta

  4
• The relation between the zeros and the coefficients is verified
• The concept of using factorization to find zeros and then verifying the relation between zeros and coefficients is explained
• The next question is introduced with the roots X = 2/3 and X = -3
• To find the coefficients a and b, the roots are substituted into the polynomial equation ax^2 + 7x + b = 0
• Substitution in a polynomial equation

  Substitute values in the polynomial equation ax squared plus 7x plus B equals 0 to find the values of a and B
• If a is a zero of a polynomial P X, then X minus a divides P X or X minus a is a factor of P X
• If 2 is a zero for polynomial P X, then X minus 2 divides P X or X minus 2 is a factor of P X
• Finding zeros of a polynomial

  Equating the polynomial to 0 and using the given zeros to find factors of the polynomial
• Root 3 and -root 3 are zeros of the polynomial 2x cubed plus x squared minus 6x minus 3
• The polynomial can be written as 2x cubed plus x squared minus 6x minus 3
• The polynomial can be factored as (x square - 3)(2x + 1)
• The other zero of the polynomial is X equal to -1/2
• If the polynomial X to the power 4 plus 2x cubed plus 8x squared plus 12x plus 18 is divided by another polynomial
• Dividing polynomials

  Dividing one polynomial by another to find the other zero
• When a polynomial is divided by another polynomial, the remainder comes out to be of the form px + Q
• To find the values of P and Q, compare the remainder with the form px + Q
• The quotient of dividing X^4 + 2x^3 + 8x^2 + 12x + 18 by X^2 + 5 is X^2 + 2x + 3
• The remainder of dividing X^4 + 2x^3 + 8x^2 + 12x + 18 by X^2 + 5 is 2x + 3
• Comparing the remainder with the form px + Q

  • P = 2
  • Q = 3
• The zeros of the polynomial x^2 + x - P(P + 1) are P and -P + 1
• The zeros of the polynomial ax^2 - 6x - 6 are 4
• To find the value of a, the product of the zeros of the polynomial ax^2 - 6x - 6 is given as 4