Basic numeracy-basic-algebra

1,501 views

Published on

Basic numeracy, basic algebra

Published in: Education
0 Comments
1 Like
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total views
1,501
On SlideShare
0
From Embeds
0
Number of Embeds
21
Actions
Shares
0
Downloads
60
Comments
0
Likes
1
Embeds 0
No embeds

No notes for slide

Basic numeracy-basic-algebra

  1. 1. Click Here to Join Online Coaching Click Here www.upscportal.com Basic Numeracy
  2. 2. www.upscportal.com Click Here to Join Online Coaching Click Here Basic Algebra Polynomial An expression in term of some variable(s) is called a polynomial. For example f(x) = 2x – 5 is a polynomial in variable x g(y) = 5y2 – 3y + 4 is a polynomial in variable y Note that the expressions like etc. are not polynomials. Thus, a rational x integral function of ‘x’ is said to be a polynomial, if the powers of ‘x’ in the terms of the polynomial are neither fractions nor negative. Thus, an expression of the form
  3. 3. www.upscportal.com Click Here to Join Online Coaching Click Here f(x) = an xn + an–1xn–1 + … + alx + a0 is called a polynomial in variable x where n be a positive integer and a0, al, ...,an be constants (real numbers). Degree of a Polynomial The exponent of the highest degree term in a polynomial is known as its degree. For example f(x) = 4x-3/2 is a polynomial in the variable x of degree 1. p(u) = 3u3 + u2 + 5u – 6 is a polynomial in the variable u of degree 3. q(t) = 5 is a polynomial of degree zero and is called a constant polynomial. Linear Polynomial A polynomial of degree one is called a linear polynomials. In general f(x) = ax + b, where a ≠ 0 is a linear polynomial.
  4. 4. www.upscportal.com Click Here to Join Online Coaching Click Here For example f(x) = 3x – 7 is a binomial as it contains two terms. g(y) = 8y is a monomial as it contains only one terms. Quadratic Polynomials A polynomial of degree two is called a quadratic polynomials. In general f(x) = ax2 + bx + c, where a ≠ 0 is a quadratic polynomial. For example f(x) = x2 – 7x + 8 is a trinomial as it contains 3 terms g(y) = 5x2 – 2x is a binomial as it contains 2 terms p(u) = 9x2 is a monomial as it contains only 1 term Cubic Polynomial A polynomial of degree 3 is called a cubic polynomial in general. f(x) = ax3 + bx3 + cx + d, a ≠ 0 is a cubic polynomial. For example f(x) = 2x3 – x2 + 8x + 4
  5. 5. www.upscportal.com Click Here to Join Online Coaching Click Here Biquadratic Polynomial A fourth degree polynomial is called a biquadratic polynomial in general. f(x) = ax4 + bx3 + cx2 + dx + e, a ≠ 0 is a bi quadratic polynomial. Zero of a Polynomial A real number a is a zero (or root) of a polynomial f(x), if f (a) = 0 For example If x = 1 is a root of the polynomial 3x3 – 2x2 + x – 2, then f(l)= 0 f(x) = 3x3 – 2x2 + x – 2, f(1) = 3 × 13 – 2 × 12 + 1 – 2 = 3 – 2 + 1 – 2 = 0, As f(1) = 0 x = 1 is a root of polynomial f(x) (1) A polynomial of degree n has n roots. (2) A linear polynomial of f(x) = ax + b, a ≠ 0 has a unique root given by x = -b/a
  6. 6. www.upscportal.com Click Here to Join Online Coaching Click Here (3) Every real number is a root of the zero polynomial. (4) A non-zero constant polynomial has no root. Remainder Theorem Let f(x) be a polynomial of a degree greater than or equal to one and a be any real number, if f(x) is divisible by (x – a), then the remainder is equal to f(a). Example 1: Find the remainder when f(x) = 2x3 – 13x2 + 17x + 10 is divided by x – 2. Solution. When f(x)is divided by x – 2, then remainder is given by f(2) = 2(2)3 – 13(2)2 + 17(2) + 10 = 16 – 52 + 34 + 10 = 8 Thus, on dividing f(x) = 22 – 13x2 + 17x + 10 by x – 2, we get the remainder 8.
  7. 7. www.upscportal.com Click Here to Join Online Coaching Click Here Factor Theorem Let f(x) be a polynomial of degree greater than or equal to one and a be any real number such that f(a) = 0, then (x – a) is a factor of f(x). Conversely, if (x – a) is a factor of f(x), then f(a) = 0. Example 2: Show that x + 2 is a factor of the polygonal x2 + 4x + 4. Solution. Let f(x) = x2 + 4x + 4 (x + 2) = {x – (–2)} is a factor of f(x) if f(–2) = 0 Now, f(–2) = (–2)2 + 4(–2) + 4 = 4 – 8 + 4 = 0 Hence, x + 2, is a factor of f(x). Useful Formulae (i) (x + y)2 = x2 + y2 + 2xy (ii) (x – y)2 = x2 + y2 – 2xy (iii) (x2 – y2) = (x + y) (x – y)
  8. 8. www.upscportal.com Click Here to Join Online Coaching Click Here (iv) (x + y)3 = x3 + y3 + 3xy(x + y) (v) (x – y)3 = x3 – y3 – 3xy(x – y) (vi) (x3 + y3) = (x + y) (x2 + y2 – xy) (vii) (x3 – y3) = (x – y) (x2 + y2 + xy) (viii) (x + y + z)2 = x2 + y2 + z2 + 2(xy + yz + zx) (ix) (x3 + y3 + z3 – 3xyz) = (x + y + z) (x2 + y2 + z2 – xy – yz – zx) (x) If x + y + z = 0, then x3 + y3 + z3 = 3xyz Also, (i) (xn – an ) is divisible by (x – a) for all values of n. (ii) (xn + an ) is divisible by (x + a) only when n is odd. (iii) (xn– an ) is divisible by (x + a) only for even values of n. (iv) (xn + an) is never divisible by (x – a).
  9. 9. www.upscportal.com Click Here to Join Online Coaching Click Here Example 3 : Factorise 216x3 – 125y3 Solution. 216x3 – 125y3 = (6x)3 – (5y)3 [using x3– y3 = (x – y) (x2 +y2 + xy)] = (6x – 5y) [(6x)2 + (5y)2 + (6x) (5y)] = (6x – 5y) (36x2 + 25y2 + 30xy) Example 4: Divide– 14x2 – 13x + 12 by 2x + 3. Solution.
  10. 10. www.upscportal.com Click Here to Join Online Coaching Click Here Maximum and Minimum Value of a Polynomial Let f(x) be a polynomial. Then, f(x) has locally maxima of minima values at a, if f(a) = 0. If f(a) > 0, then f(x) has minimum value at x = a. If f(a)< 0, then f(x) has maximum value at x = a. Example 5: Which of the following is not a polynomial? (a) 5x2 – 4x + 1 (c) x – 2/5 Solution. (a) 5x2 – 4x + 1 is a quadratic polynomial in one variable. (b) is not a polynomial as it does not contain an integral power of x. (c) is not a polynomial as it does not contain an integral power of x
  11. 11. www.upscportal.com Click Here to Join Online Coaching Click Here Click Here to Join Online Coaching: http://upscportal.com/civilservices/courses Click Here to Buy Study Material in Hard Copy: http://upscportal.com/civilservices/study-kit

×