1. Geometrical Meaning of the Zeroes of a Polynomial
Any polynomial in one variable can be represented by a graph. A linear polynomial can be represented by a line, a quadratic polynomial by a parabola, a cubic polynomial by a cubic curve, etc.
The points at which these graphs meet the x-axis, i.e., the points at which the polynomial function is equal to zero, represent the zeroes or roots of the polynomials.
Geometrical meaning of the zeroes of a polynomial
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.
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 this case, the shape of the graph is a parabola.
The shape of the parabola of any quadratic polynomial of the form ax2 + bx + c, a ≠ 0 depends on a.
If a > 0, then the parabola opens upwards.
If a 0, hen the parabola opens downwards.
The graph of a cubic polynomial intersects the x-axis at a maximum of three points. A cubic polynomial has a maximum of 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.
2. Relationship between Zeroes and Coefficient of a Polynomial
Polynomials can be linear (x), quadratic (x2), cubic (x3) and so on, depending on the highest power of the variable.
The number of zeroes of a polynomial is equal to the degree of the polynomial, and there is a well-defined mathematical relationship between the zeroes and the coefficients. Mathematically, if p(x) is a polynomial in x, and if k is any real number, then the value obtained by replacing x by k in p(x) is called the value of p(x) at x = k and is denoted by p(k).
Here, the real number k is said to be a zero of the polynomial of p(x), if p(k) = 0.
Simply put, the zeroes of a polynomial function are the solutions to the equation you get, when you set the polynomial equal to zero.
Let us understand the difference between zeros and roots in a polynomial equation.
- A zero is a value for which a polynomial is equal to zero.
- When you set a polynomial equal to zero, then you have a polynomial equation where the equations roots are same as the polynomial’s zeroes.
- A root is a value for which a polynomial equation is true.
- Example: The polynomial x-5 has one zero, that is x = 5. And the polynomial equation x-5 = 0 has one root, that is, x = 5.
The number of zeroes of a polynomial is equal to the degree of the polynomial, and there is a well-defined mathematical relationship between the zeroes and the coefficients.
Linear Polynomial
Any equation of the first degree is known as a linear equation. It is an equation of the form ax+b = 0, where and b are constants and x is a variable.
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Quadratic Polynomial
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- Product of the toots = span id="MathJax-Element-8-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0" data-mathml="=">αβ=αβ= span id="MathJax-Element-9-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0" data-mathml="ca">caca = span id="MathJax-Element-10-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0" data-mathml="(coefficientofx)(coefficientofx2)">(coefficientofx)(coefficientofx2)(coefficientofx)(coefficientofx2)
Cubic Polynomial
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In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised version of the familiar arithmetic technique called long division. It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones.
Polynomial long division can be used to find the equation of the line that is tangent to the graph of the function defined by the polynomial P(x) at a particular point x = r.
Division Algorithm
Let f(x), g(x), q(x) and r(x) are polynomials then the division algorithm for polynomials states that “If f(x) and g(x) are two polynomials such that degree of f(x) is greater than degree of g(x) where g(x) ≠ 0, then there exists unique polynomials q(x) and r(x) such that f(x) = g(x).q(x) + r(x) where r(x) = 0 or degree of r(x) < degree of g(x).
In other words, Dividend = divisor x quotient + remainder