Notes On Geometrical Meaning of Zero of Polynomials - CBSE Class 10 Maths
Polynomial A polynomial is an algebraic expression of the form p(x) = an xn + an-1 xn-1 +........+ a1 x1 + a0, where an, an-1, a1, a0 are real numbers and an≠ 0 is called a polynomial. where an xn , an-1 xn-1 +........, a1 x1 , a0, are the terms of a polynomial and an , an-1 ,........, a1, a0 are their coefficients. Coefficient The real number that precedes the variable is called the coefficient. A polynomial involving one variable is called a polynomial in one variable. Degree of the Polynomial The exponent of the highest power of the variable of a polynomial is called the degree of the polynomial. Based on its degree, a polynomial can be called as linear polynomial, quadratic polynomial, cubic  polynomial and so on. Zero Polynomial The polynomial with all the coefficients as zeros is called a zero polynomial. Constant Polynomial A polynomial with a single term of a real number is called a constant polynomial. Linear Polynomial A polynomial of degree one is called a first-degree or linear polynomial. The general form of a linear polynomial is ax + b, where a and b are real numbers and a ≠ 0. Quadratic Polynomial A polynomial of degree two is called a second degree or quadratic polynomial. The general form of a quadratic polynomial is ax2 + bx + c, where a,b and c are real numbers and a ≠ 0. Cubic Polynomial A polynomial of degree three is called a third-degree or cubic polynomial. The general form of a cubic polynomial is ax3 + bx2 + cx + d, where a,b,c and d are real numbers and a ≠ 0. BI-Quadratic Polynomial A polynomial of degree four is called a fourth-degree or biquadratic polynomial. The general form of a quadratic polynomial is ax4 + bx3 + cx2 + dx + e, where a,b,c,d and e are real numbers and a ≠ 0. Value of a Polynomial The value of a polynomial p(x) when x = k (k is a real number) is the value obtained by substituting x as k. It is denoted by p(k). Zero of the Polynomial The zero of the polynomial is defined as any real value of x, for which the value of the polynomial becomes zero. A real number k is a zero of a polynomial p(x), if p(k) = 0. 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 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 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. Geometrically a quadratic polynomial can have either two distinct zeroes or two equal zeroes or no zero. This indicates that a quadratic polynomial has almost 2 zeroes. The graph of a cubic polynomial intersects the x-axis at 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.

#### Summary

Polynomial A polynomial is an algebraic expression of the form p(x) = an xn + an-1 xn-1 +........+ a1 x1 + a0, where an, an-1, a1, a0 are real numbers and an≠ 0 is called a polynomial. where an xn , an-1 xn-1 +........, a1 x1 , a0, are the terms of a polynomial and an , an-1 ,........, a1, a0 are their coefficients. Coefficient The real number that precedes the variable is called the coefficient. A polynomial involving one variable is called a polynomial in one variable. Degree of the Polynomial The exponent of the highest power of the variable of a polynomial is called the degree of the polynomial. Based on its degree, a polynomial can be called as linear polynomial, quadratic polynomial, cubic  polynomial and so on. Zero Polynomial The polynomial with all the coefficients as zeros is called a zero polynomial. Constant Polynomial A polynomial with a single term of a real number is called a constant polynomial. Linear Polynomial A polynomial of degree one is called a first-degree or linear polynomial. The general form of a linear polynomial is ax + b, where a and b are real numbers and a ≠ 0. Quadratic Polynomial A polynomial of degree two is called a second degree or quadratic polynomial. The general form of a quadratic polynomial is ax2 + bx + c, where a,b and c are real numbers and a ≠ 0. Cubic Polynomial A polynomial of degree three is called a third-degree or cubic polynomial. The general form of a cubic polynomial is ax3 + bx2 + cx + d, where a,b,c and d are real numbers and a ≠ 0. BI-Quadratic Polynomial A polynomial of degree four is called a fourth-degree or biquadratic polynomial. The general form of a quadratic polynomial is ax4 + bx3 + cx2 + dx + e, where a,b,c,d and e are real numbers and a ≠ 0. Value of a Polynomial The value of a polynomial p(x) when x = k (k is a real number) is the value obtained by substituting x as k. It is denoted by p(k). Zero of the Polynomial The zero of the polynomial is defined as any real value of x, for which the value of the polynomial becomes zero. A real number k is a zero of a polynomial p(x), if p(k) = 0. 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 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 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. Geometrically a quadratic polynomial can have either two distinct zeroes or two equal zeroes or no zero. This indicates that a quadratic polynomial has almost 2 zeroes. The graph of a cubic polynomial intersects the x-axis at 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.

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