Portfolio of ch polynomial class 9
Answers
An expression p(x) = a0xn + a1xn–1 + a2xn–2 + ... an is a polynomial Where a0, a1, .......... an are real numbers and n is non-negative integer.
• Degree of a polynomial is the greatest exponent of the variable in the polynomial.
• Constant polynomial is a polynomial of degree zero. The constant plynomial f(x) = 0 is called zero polynomial.
• Degree of zero polynomial is not defined.
• A polynomial of degree one is called a linear polynomil e.g. ax + b, where a ≠ 0.
• A polynomial of degree two is called a quadratic polynomial e.g. ax2 + bx + c where a ≠ 0.
• A polynomial of degree 2 is called a cubic polynomial e.g. px3 + qx2 + rx + s, p ≠ 0.
• A polynomial of degree 4 is called a biquadratic polynomial e.g. px4 + qx3 + rx2 + sx + t, p ≠ 0.
• Value of a polynomial p(x) at x – a is p(a).
• Zero of a polynomial p(x) is a number ‘a’ such that p(a) = 0.
REMAINDER THEOREM
• Let p(x) is a polynomial of degree greater than or equal to 1 and a is any real number, if p(x0 is divided by the linear polynomial x – a then the remainder is p(a).
FACTOR THEOREM
• If p(x) is a polynomial of degree x �d 1 and a is any real number then.
(i) x – a is a factor of p(x) if p(a) = 0.
(ii) p(a) = 0 if (x – a) is a factor of p(x).
ALGEBRIC IDENTITIES
• (x + y)2 = x2 + 2xy + y2
• (x – y)2 = x2 – 2xy + y2
• x2 – y2 = (x + y)(x – y)
• (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
• (x + a)(x + b) = x2 + (a + b)x + ab
• (x + y)3 = x3 + y3 + 3xy(x + y)
• (x – y)3 = x3 – y3 – 3xy(x – y)
• x3 + y3 + z3 – 3xyz = [(x + y + z) (x2 + y2 + z2 – xy – yz – zx)]
• If x + y + z = 0, then x3 + y3 + z3 = 3xyz.
• x3 – y3 = (x + y) (x2 + xy + y2)
• x3 – y3 = (x – y) (x2 + xy + y2)
DEGREE OF A POLYNOMIAL
• The exponent of the term with the highest power in a polynomial is known as its degree. f(x) = 8x3 – 2x2 + 8x – 21 and g(x) = 9x2 – 3x + 12 are polynomials of degree 3 and 2 respectively.
ZEROS OF A POLYNOMIAL
• Value of polynomial: The value of a polynomial f(x) at x = c is obtained by substituting x = c in the given polynomial and is denoted by f(c).
• Zero or root: A real number c is a zero of the polynomial f(x) = a0xn + a1xn–1 + ... + an, if f(c) = 0. ⇒ a0cn + a1cn–1 + a2cn–2 + ... + an = 0
Answer:
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