Basic numeracy-basic-algebra

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Basic Numeracy

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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

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f(x) = an xn + an–1x

n–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.

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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

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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

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(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.

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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)

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(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).

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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.

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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

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