Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy.

Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our Privacy Policy and User Agreement for details.

Like this presentation? Why not share!

- Basic Algebra Ppt 1.1 by Cincinnati State 14315 views
- Number systems / Algebra by indianeducation 397 views
- Basic numeracy-sequences-and-series by iasexamportal.com 421 views
- First course in algebra and number ... by Educationtempe75 20 views
- Basic algebra for entrepreneurs by Dr. Trilok Kumar ... 2505 views
- Power point number by nishamolnnn 203 views

1,501 views

Published on

Basic numeracy, basic algebra

Published in:
Education

No Downloads

Total views

1,501

On SlideShare

0

From Embeds

0

Number of Embeds

21

Shares

0

Downloads

60

Comments

0

Likes

1

No embeds

No notes for slide

- 1. Click Here to Join Online Coaching Click Here www.upscportal.com Basic Numeracy
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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

No public clipboards found for this slide

×
### Save the most important slides with Clipping

Clipping is a handy way to collect and organize the most important slides from a presentation. You can keep your great finds in clipboards organized around topics.

Be the first to comment