What is a Polynomial?
§ A polynomial is an expression consisting of variables and coefficients, that
involves only the operations of addition, subtraction, multiplication, and
non-negative integer exponents.
An example of a polynomial of a single variable, x, is x² − 4x + 7, which is a
quadratic polynomial.
§ Expressions like 1/x - 1, √x+2, x¾ + x² - 7x + 3 etc., are not
polynomials.
3r2[]gf3;ge4, whil,,,,
involves only the operations of addition, subtraction, multiplication, and
non-negative integer exponents.
An example of a polynomial of a single variable, x, is x² − 4x + 7, which is a
quadratic polynomial.
§ Expressions like 1/x - 1, √x+2, x¾ + x² - 7x + 3 etc., are not
polynomials.
3r2[]gf3;ge4, whil,,,,
Types of polynomials
§ An algebraic expression containing two
unlike terms is called a binomial.
§ An algebraic expression containing
three unlike terms is called a trinomial.
Degrees of a polynomial
§ In a polynomial p(x), the highest power of x in p(x) is called the degree of the polynomial p(x).
§ A polynomial of degree one is called a linear polynomial. It is of
the form ax + b where a, b are real numbers and a≠0.
Examples: 5x - 3, 2x etc.
§ A polynomial of degree two is called a quadratic polynomial. The
name ‘quadratic’ has been derived from the word ‘quadrate’,
which means ‘square’. It is of the form ax² + bx + c .
where a, b, c are real numbers and a≠0.
Examples: 2x² + x - 1, 1 - 5x + x² etc.
§ A polynomial of degree three is called a cubic polynomial and
has the general form ax³+ bx² + cx +d, where a, b, c and d
are real numbers and a ≠ 0.
Examples : √3x³ - x + √5, x³ - 1 etc.
§ The graph of a polynomial p(x) of degree n can intersects
or touch the x-axis at atmost n points.
§ A polynomial of degree n has at most n distinct real zeroes.
§ A polynomial of degree one is called a linear polynomial. It is of
the form ax + b where a, b are real numbers and a≠0.
Examples: 5x - 3, 2x etc.
§ A polynomial of degree two is called a quadratic polynomial. The
name ‘quadratic’ has been derived from the word ‘quadrate’,
which means ‘square’. It is of the form ax² + bx + c .
where a, b, c are real numbers and a≠0.
Examples: 2x² + x - 1, 1 - 5x + x² etc.
§ A polynomial of degree three is called a cubic polynomial and
has the general form ax³+ bx² + cx +d, where a, b, c and d
are real numbers and a ≠ 0.
Examples : √3x³ - x + √5, x³ - 1 etc.
§ The graph of a polynomial p(x) of degree n can intersects
or touch the x-axis at atmost n points.
§ A polynomial of degree n has at most n distinct real zeroes.
Geometrical meaning of the zeroes and the Geometrical representations of the polynomial
Zeroes of a polynomial : A real number k is called a zero of polynomial p(x) if p(k) = 0.The graph of y = p(x) intersects
the X- axis.
Geometrical Meaning of the Zeroes of a Polynomial: The zero of the polynomial is the x-coordinate of the point, where the graph intersects the x-axis. If a polynomial p(x) intersects the x-axis at (k,0) , then k is the zero of the polynomial.
the X- axis.
Geometrical Meaning of the Zeroes of a Polynomial: The zero of the polynomial is the x-coordinate of the point, where the graph intersects the x-axis. If a polynomial p(x) intersects the x-axis at (k,0) , then k is the zero of the polynomial.
§ A linear polynomial ax + b, a ≠ 0, the graph of y = ax + b is a
straight line which intersects the x-axis at exactly one point. § The graph of a linear polynomial intersects the x-axis at a maximum of one point. Therefore, a linear polynomial has a maximum of one zero. Here, the graph cuts the x-axis at exactly one point, i.e., at two coincident points. So, the two points A and A′ of Case (I) coincide here to become one point A. § The graph of a quadratic polynomial intersects the x-axis at a maximum of two points. Therefore, a quadratic polynomial can have a maximum of two zeroes. In case of a quadratic polynomial, the shape of the graph is a parabola. The shape of the parabola of a quadratic polynomial ax² + bx + c, a ≠ 0 depends on a. § If , a > 0 then the parabola opens upwards. § If , a < 0 then the parabola opens downwards. § Here, the graph cuts x-axis at two distinct points A and A′. The x-coordinates of A and A′ are the two zeroes of the quadratic polynomial ax + bx + c. § The quadratic polynomial can have no zero. § The graph is either completely above the x-axis or completely below the x-axis. So, it does not cut the x-axis at any point. § So, you can see geometrically that a quadric polynomial can have either two distinct zeroes or two equal zeroes (i.e., one zero)or no zero. § The graph of a cubic polynomial intersects the x -axis at maximum of three points. A cubic polynomial can have atmost three zeroes. § In general, an nth-degree polynomial intersects the x-axis at a maximum of n points. Therefore, an nth-degree polynomial has a maximum of n zeroes. |
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Relationship between Zeroes and coefficients of a Polynomial
§ A real
number k is said to be a zero of a polynomial p(x), if on substituting x=k in a
polynomial p(x), we get p(k)=0.
§ The zero of the linear polynomial ax + b is -b/a .
§ General form of quadratic polynomial is ax²+bx+c. There are two zeroes of quadratic
polynomial.
[α,β and γ are Greek letters pronounced as 'alpha', 'beta' and 'gamma' respectively].
§ The zero of the linear polynomial ax + b is -b/a .
§ General form of quadratic polynomial is ax²+bx+c. There are two zeroes of quadratic
polynomial.
[α,β and γ are Greek letters pronounced as 'alpha', 'beta' and 'gamma' respectively].
Division Algorithm for Polynomials
The division algorithm for states that any polynomial p(x), g(x), q(x) and r(x) are polynomials then “If p(x) and g(x) are two polynomials such that degree of p(x) is greater that degree of g(x) where g(x) ≠ 0, then there exists unique polynomials q(x) and r(x) such that p(x) = g(x)*q(x) + r(x),
where, r(x) = 0 or degree of r (x) < (less than) degree of g (x)”.
This result is known as the Division Algorithm for polynomials.
Dividend = Divisor x Quotient + Remainder.
§ Factor Theorem: If a is zero of a polynomial p(x) then (x – a) is a factor of p(x).
Example: x²-3x-4
p(4) = (4)² -3(4) - 4 = 16-12-4 = 0
so (x - 4) must be a factor of x² - 3x - 4
§ Polynomial Long Division:
Example: 2x²-5x-1 divided by x-3
After dividing we get the answer 2x+1, but there is a remainder of 2.
§ The Remainder Theorem: When we divide a polynomial f(x) by x- c we get:
f(x) = (x-c)·q(x) + r(x)
But r(x) is simply the constant r (remember? when we divide by (x-c) the remainder is a constant) ....
so we get this:
f(x) = (x-c)·q(x) + r
Now see what happens when we have x equal to c:
f(c) = (c-c)·q(c) + r
f(c) = (0)·q(c) + r
f(c) = r
So we get the Remainder theorem as follows:
When we divide a polynomial f(x) by x-c the remainder equals f(c).
where, r(x) = 0 or degree of r (x) < (less than) degree of g (x)”.
This result is known as the Division Algorithm for polynomials.
Dividend = Divisor x Quotient + Remainder.
§ Factor Theorem: If a is zero of a polynomial p(x) then (x – a) is a factor of p(x).
Example: x²-3x-4
p(4) = (4)² -3(4) - 4 = 16-12-4 = 0
so (x - 4) must be a factor of x² - 3x - 4
§ Polynomial Long Division:
Example: 2x²-5x-1 divided by x-3
- f(x) is 2x²-5x-1
- g(x) is x-3
After dividing we get the answer 2x+1, but there is a remainder of 2.
- q(x) is 2x+1
- r(x) is 2
§ The Remainder Theorem: When we divide a polynomial f(x) by x- c we get:
f(x) = (x-c)·q(x) + r(x)
But r(x) is simply the constant r (remember? when we divide by (x-c) the remainder is a constant) ....
so we get this:
f(x) = (x-c)·q(x) + r
Now see what happens when we have x equal to c:
f(c) = (c-c)·q(c) + r
f(c) = (0)·q(c) + r
f(c) = r
So we get the Remainder theorem as follows:
When we divide a polynomial f(x) by x-c the remainder equals f(c).
Flow Chart of Polynomials
Real-life applications of Polynomials
§ In the real world, algebra and calculus concepts are
essential to career paths in the areas of construction,
architecture, aerospace and financial planning.
§ The most obvious of these are mathematicians, but they can also be used in fields ranging from construction to
meteorology.
architecture, aerospace and financial planning.
§ The most obvious of these are mathematicians, but they can also be used in fields ranging from construction to
meteorology.