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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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