Polynomials | Revision Notes

Polynomials | Revision Notes

Polynomials

  • An expression of the form p(x) = a0 + a1x + a2x2 + a3x3 + ... + anxn , where p(x), is called a polynomial in x of degree n.

Here, a0, a1, a2, a3, ... a are real numbers and each power of x is a non-negative integer.

  • 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.
  • A polynomial of degree 1 is called a linear polynomial. A linear polynomial is of the form p(x) = ax + b, where a ≠ 0.
  • A polynomial of degree 2 is called a quadratic polynomial. A quadratic polynomial is of the form p(x) = ax2 + bx + c, where a ≠ 0,.
  • A polynomial of degree 3 is called a cubic polynomial. A cubic polynomial is of the form p(x) = ax3 + bx2 + cx + d, where a ≠ 0.
  • A polynomial of degree 4 is called a biquadratic polynomial. A biquadratic polynomial is of the form p(x) = ax4 + bx3 + cx2 + dx + e, where a ≠ 0,.
  • If p(x) is a polynomial in x and if α is any real number, then the value obtained by putting x = α in p(x) is called the value of p(x) at x = α . The value of p(x) at x = α is denoted by p(α) .
  • A real number α is called a zero of the polynomial p(x), if p(α) = 0.
  • A polynomial of degree n can have at most n real zeroes.
  • Geometrically the zeroes of a polynomial p(x) are the x-coordinates of the points, where the graph of p(α) = 0. intersects x-axis.
  • Zero of the linear polynomial ax + b is 

\frac{-b}{a}=\frac{- \emph{constant term}}{\emph{coefficient of }x}

  • If α and β are the zeroes of a quadratic polynomial p(x) = ax2 + bx + c, a ≠ 0,, then

\alpha +\beta =\frac{-b}{a}=\frac{- \text{coefficient of }x}{\text{coefficient of }x^2}

 

\alpha \beta =\frac{c}{a}=\frac{ \text{constant term }}{\text{coefficient of }x^2}

  • If α , β and γ are the zeroes of a cubic polynomial p(x) = ax3 + bx2 + cx + d, a ≠ 0, then

\alpha +\beta + \gamma =\frac{-b}{a}=\frac{- \text{coefficient of }x^2}{\text{coefficient of }x^3}

 

\alpha\beta +\beta\gamma + \gamma\alpha =\frac{c}{a}=\frac{\text{coefficient of }x}{\text{coefficient of }x^3}

 

\alpha \beta \gamma =\frac{-d}{a}=\frac{- \text{constant term }}{\text{coefficient of }x^3}

  • A quadratic polynomial whose zeroes are α , β is given by p(x) = x2 – (α + β)x + αβ = x2 – (sum of the zeroes) x + product of the zeroes.
  • A cubic polynomial whose zeroes are α, β , γ is given by p(x) = x3 – (α + β + γ) x2 + (αβ + βγ + γα)x + αβγ = x3 – (sum of the zeroes) x2 + (sum of the products of the zeroes taken two at a time) x – product of the zeroes.
  • 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).

Important Questions

  1. For what value of k, (–4) is a zero of the polynomial x2 – x – (2k + 2)?
  2. If the zeroes of the polynomial x3 – 3x2 + x + 1 are a – b, a and a + b, fnd the values of a and b.
  3. Find a quadratic polynomial whose zeroes are 3 + √5 and 3–√5.
  4. α, β are the roots of the quadratic polynomial p(x) = x2 – (k – 6) x + (2k + 1). Find the value of k, if α + β = αβ .
  5. On dividing x3 – 3x2 + x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and –2x + 4 respectively. Find g(x).
  6. Find the zeroes of 4√5x2 + 17x + 3√5 and verify the relation between the zeroes and coeffcients of the polynomial.
  7. Find the zeroes of the quadratic polynomial p(x) = x2 – (√3 +1)x + 3 and verify the relationship between the zeroes and its coeffcients.
  8. If α and β are the zeroes of the quadraticpolynomial f(x) = x2 – px + q, prove that
    \frac{\alpha^2 }{\beta^2 }+\frac{\beta^2 }{\alpha^2 }=\frac{p^4}{q^2}-\frac{4p^2}{q}+2.
  9. If α and β are the zeroes of the quadraticpolynomial p(s) = 3s2 – 6s + 4, fnd the value of
    \frac{\alpha}{\beta}+\frac{\beta}{\alpha}+2\left ( \frac{1}{\alpha }+\frac{1}{\beta } \right )+3\alpha \beta.
  10. If α and β are zeroes of the quadratic polynomial f(x) = kx2 + 4x + 4 such that α2 + β2 = 24, find the value of k.
  11. If one zero of the quadratic polynomial f(x) = 4x2 – 8kx – 9 is negative of the other, find the value of k.
  12. If the sum of the zeroes of the quadratic polynomial f(t) = kt2 + 2t + 3k is equal to their product, find the value of k.
  13. If the square of the difference of the zeroes of the quadratic polynomial f(x) = x2 + px + 45 is equal to 144, find the value of p.
  14. If one zero of the polynomial (a2 + 9)x2 + 13x + 6a is reciprocal of the other, find the value of a.
  15. If α, β, γ are zeroes of the polynomial 6x3 + 3x2 – 5x + 1, then fnd the value of α–1 + β–1 + γ–1.
  16. Find a cubic polynomial with the sum, sum of the products of its zeroes taken two at a time and product of its zeroes as 3, –1 and –3 respectively.
  17. If α, β are the zeroes of the polynomial 21y2 – y – 2, find a quadratic polynomial whose zeroes are 2α and 2β.
  18. If α, β are the zeroes of the polynomial 6y2 – 7y + 2, find a quadratic polynomial whose zeroes are \frac{1}{\alpha } \text{ and } \frac{1}{\beta}.
  19. If α and β are the zeroes of the quadratic polynomial f(x) = x2 + px + q, form a polynomial whose zeroes are (α + β)2 and (α – β)2.
  20. If α and β are the zeroes of the polynomial f(x) = x2 – 2x + 3, find a polynomial whose zeroes are α + 2 and α + β.
  21. Divide 30x4 + 11x3 – 82x2 – 12x + 48 by (3x2 + 2x – 4) and verify the result by division algorithm.
  22. If α and β are zeroes of the quadratic polynomial x2 – 6x + a; fnd the value of a if 3α + 2β = 20.
  23. Find the value of p for which the polynomial x3 + 4x2 – px + 8 is exactly divisible by x – 2.
  24. If the remainder on division of x3 + 2x2 + kx + 3 by x – 3 is 21, fnd the quotient and the value of k. Hence, fnd the zeroes of the cubic polynomial x3 + 2x2 + kx – 18.
  25. If the polynomial x4 – 6x3 + 16x2 – 25x + 10 is divided by another polynomial x2 – 2x + k, the remainder comes out to be (x + a), fnd the values of k and a.
  26. Find all other zeroes of the polynomial p(x) = 2x3 + 3x2 – 11x – 6, if one of its zero is –3.
  27. Find all the zeroes of the polynomial 2x4 – 3x3 – 5x2 + 9x – 3, it being given that two of its zeros are √3 and –√3.
  28. Find all the zeroes of the polynomial 2x4 – 10x3 + 5x2 + 15x – 12, if it is given that two of its zeroes are \small \sqrt{\frac{3}{2}} \text{ and} -\sqrt{\frac{3}{2}} .
  29. Find all the zeroes of the polynomial x4 + x3 – 34x2 – 4x + 120, if two of its zeroes are 2 and –2.
  30. If two zeroes of p(x) = x4 – 6x3 – 26x2 + 138x – 35 are 2 ± √3 , fnd the other zeroes.
  31. If the polynomial 6x4 + 8x3 – 5x2 + ax + b is exactly divisible by the polynomial 2x2 – 5, then fnd the values of a and b.
  32. What must be added to the polynomial P(x) = 5x4 + 6x3 – 13x2 – 44x + 7 so that the resulting polynomial is exactly divisible by the polynomial Q(x) = x2 + 4x + 3 and the degree of the polynomial to be added must be less than degree of the polynomial Q(x).
  33. What must be subtracted from the polynomial f(x) = x4 + 2x3 – 13x2 – 12x + 21 so that the resulting polynomial is exactly divisible by g(x) = x2 – 4x + 3?