Facts That Matter- Polynomials - Class 10 PDF Download

Facts that Matter

• Polynomial
  An algebraic expression of the form p(x) = a₀ + a₁ x + a₂ x² + a₃ x³ + ...... + a_n xⁿ, in which the variables involved have only non-negative integral exponents, is called a polynomial in x of degree n.

Note: In the polynomial a₀ + a₁ x + a₂ x² + .... + a_n xⁿ
1. a_n ≠ 0
2. a₀, a₁ x¹, a₂ x², a₃ x³, ..... a_{n - 1} x^{n - 1}, a_nxⁿ are terms.
3. a₀, a₁, a₂, ..... a_{n - 1}, a_n are the co-efficients of x⁰, x¹, x², ....., x^n-1, xⁿ respectively.

• Degree of a Polynomial
  The highest power of the variable in a polynomial is called its degree. For example,
  p(x) = 5x + 3 is a polynomial in x of degree 1.
  p(y) = 3y² + 4y - 4 is a polynomial in y of degree 2.
• Linear Polynomial
  A polynomial of degree 1 is called a linear polynomial. A linear polynomial is generally written as ax + b (a ≠ 0), where a, b are real coefficients.
• Quadratic Polynomial
  A polynomial of degree 2 is called a quadratic polynomial. A quadratic polynomial is generally written as ax² + bx + c (a ≠ 0), where a, b and c are real coefficients.
• Cubic Polynomial
  A polynomial of degree 3 is called a cubic polynomial. A cubic polynomial is generally written as ax³ + bx² + cx + d (a ≠ 0), where a, b, c and d are real coefficients.
  Examples: 
  Degree of different polynomials
  

• Value of a Polynomial at a Given Point
  If p (x) is a polynomial in x and ‘a’ is a real number. Then the value obtained by putting
  x = a in p (x) is called the value of p (x) at x = a.
  Example:
  Let p(x) = 5x² - 4x + 2 then its value at x = 2 is given by
  p (2) = 5 (2)² - 4 (2) + 2
  = 5 (4) - 8 + 2
  = 20 - 8 + 2
  = 14
  Thus, the value of p(x) at x = 2 is 14.
• Zeroes of a Polynomial
  A real number ‘a’ is said to be a zero of the polynomial p (x), if p (a) = 0.
  Example:
  Let p (x) = x² - x - 2
  Then p (2) = (2)² - (2) - 2 = 4 - 4 = 0
  and p (-1) = (-1)² - (-1) - 2 = 2 - 2 = 0
  ∴ (-1) and (2) are the zeroes of the polynomial x² - x - 2.

Note:
I. A linear polynomial has at the most one zero.
II. A quadratic polynomial has at the most two zeroes.
III. In general a polynomial of degree n has at the most n zeroes.

• Geometrical Meaning of the Zeroes of a Polynomial
  First, we consider a linear polynomial p (x) = ax + b. Let ‘k’ be a zero, then
  p (k) = ak + b = 0
  ⇒ ak + b = 0
  ⇒ ak = - b
  or k =
  The graph of a linear polynomial is always a straight line. It may or may not pass through
  x-axis. In case the graph line is passing through a point on the x-axis, then the y-coordinate of that point must be zero.
  In general, for a linear polynomial ax + b = 0, (a ≠ 0), the graph is a straight line which can intersect the x-axis at exactly one point, namely,  is the zero of the polynomial ax + b.In the given figure, CD is meeting x-axis at x = -1.
  ∴ Zero of ax + b is -1.
  
  

Note: 
A zero of a linear polynomial is the x-coordinate of the point, where the graph intersects the x-axis.

• Graph of a Quadratic Polynomial
  The graph of ax² + bx + c, (a ≠0) is a curve of ∪ shape, called a parabola. If a > 0 in
  ax² + bx + c, the shape of the parabola is ∪ (opening upwards). If a < 0 in ax² + bx + c, the shape of parabola is ∩ (opening downwards).
  In the given figure, the graph of a quadratic polynomial x² - 3x - 4 is shown. It intersects x-axis at (-1, 0) and (4, 0). Therefore, its zeroes are -1 and 4. Here, a > 0, so the graph opens upwards.
  
  Whereas the following figure is a graph of the polynomial - x² + x + 6. Since it intersects the x-axis at (3, 0) and (-2, 0). Therefore, the zeroes of - x² + x + 6 are -2 and 3.
  
  Here a < 0, so the parabola opens downwards.