IMPORTANT TERMS, DEFINITIONS AND RESULTS

  1. An expression of the form
    p(x) = a_0+a_1x+a_2x^2+...a_nx^n,
    where ax^2+bx+c, is called a polynomial in  x of degree n ,
    Here, a_0,a_1,a_2,....a_n, are real numbers and each power of x is a non – negative interger.
  2. The exponent of the highest degree term in a polynomial is known as its degree. A polynomial of degree 0 is called a constant polynomial.
  3. A polynomial of degree 1 is called a linear polynomial. A linear polynomial is of the form p(x)=ax+b, where a\ne0 ,
  4. A polynomial of degree 2 is called a quadratic polynomial. A quadratic polynomial is of the form p(x) = ax^2 + bx + c, where a\ne0 .
  5. A polynomial of degree 3 is called a cubic polynomial. A cubic polynomial is of the form p(x) = ax^3 + bx^2 + cx + d, where a\ne0 .
  6. A polynomial of degree 4 is called a biquadratic polynomial. A biquadratic polynomial is of the form  p(x) = ax^4+ bx^3+ cx^2+ dx + e, where a\ne0 ,
  7. If  p(x) is a polynomial in x and if \alpha is any real number, then the value obtained by putting x=\alpha in p(x) is called the value of  p(x) at  x=\alpha. The value of p(x) at x=\alpha is denoted by p(\alpha) .
  8. A real number \alpha is called a zero of the polynomial p(x) , if p(\alpha)=0 .
  9. A polynomial of degree n can have at most n real zeroes.
  10. Geometrically the zeroes of a polynomial p(x) are the x-coordinates of the points, where the graph of p(\alpha)=0 , intersects x-axis.
  11. Zero of the linear polynomial ax+b is -\dfrac{b}{a}=\dfrac{\text{-constant term}}{\text{coefficient of x}}
  12. If \alpha and \beta are the zeroes of a quadratic polynomial p(x)=ax^2 + bx + c, a \ne 0, then
    \begin{aligned} &\alpha+\beta=-\dfrac{b}{a}=\dfrac{-\text { coefficient of } x}{\text { coefficient of } x^{2}}, \ &\alpha \beta=\frac{c}{a}=\dfrac{\text { constant term }}{\text { coefficient of } x^{2}} \end{aligned}
  13. If \alpha,\beta and \gamma are the zeroes of a cubic polynomial p(x) = ax^3 + bx^2 + cx + d, a \ne 0, then
     \begin{aligned} &\alpha+\beta+\gamma=\dfrac{-b}{a}=-\dfrac{\text { coefficient of } x^{2}}{\text { coefficient of } x^{3}} \ &\alpha \beta+\beta \gamma+\gamma \alpha=\dfrac{c}{a}=\dfrac{\text { coefficient of } x}{\text { coefficient of } x^{3}} \ &\alpha \beta \gamma=-\dfrac{d}{a}=-\dfrac{\text { constant term }}{\text { coefficient of } x^{3}} \end{aligned}
  14. A quadratic polynomial whose zeroes are \alpha,\beta is given by p(x)=x^2-(\alpha+\beta)x+\alpha\beta=x^2 โ€“ (sum of the zeroes) x + product of the zeroes.
  15. A cubic polynomial whose zeroes are \alpha,\beta,\gamma is given by p(x)=x^3-(\alpha+\beta+\gamma)x^2+(\alpha\beta+\beta\gamma+\gamma\alpha)x-\alpha\beta\gamma=x^3 โ€“ (sum of the zeroes) x^2 + (sum of the products of the zeroes taken two at a time)  x โ€“ product of the zeroes.
  16. The division algorithm states that given any polynomial p(x) and any non-zero polynomial g(x) , there are polynomial q(x) and r(x) such that p(x) = g(x)q(x) + r(x), where r(x) = 0 or degree r(x) < degree g(x).

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