What is a Polynomial What is a Polynomial Let’s learn “What is a Polynomial?” We look at the following expressions : 2x + 3 , 5x^2 + 4x + 9 , 4x^3 + 3x^2 + 7x + 3 All these expressions represent the polynomials. It will be more clear here that "What is a Polynomial and what is its degree? “ . We see that the expressions with the highest power of the variable are called the degree of the polynomial. So in the above given expressions, we have the degrees 1 , 2 and 3 respectively. Any polynomial with the degree 1 is called the linear polynomial. A polynomial with the degree 2 is called quadratic polynomial and the polynomial with the degree 3 is called the cubic polynomial. A polynomial with the degree 4 is called biquadratic polynomial. In order to find the value of the polynomial at a given point, we will replace the variable with that particular value, and the result we get is the value of the polynomial at a particular given point.
Know More About :- Green's Theorem
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So we say that if p(x) is any polynomial, then the value obtained by putting x = a in p(x) is called the value of p(x) at x = a. This value of p(x) at x = a is denoted by p(a). Now let us look at what is the zero of the polynomial? Any real number a is said t be the zero of the given polynomial p(x), when we have p(a) = 0. So this makes us clear that if we put the value of x = a, then p(x) will surely become zero. Or even we can say that the we equate p(x) = 0, to get the zero of the given polynomial. Remember here is the difference between x = 0 and p(x) = 0, we are not putting the value of x = 0, but we are equating the polynomial P(x) = 0, in order to get the value of x, which will make the polynomial p(x) = 0 The linear polynomial will have only one zero, on another hand we have maximum two zeroes of the polynomials. In the same way a cubic polynomial has at most 3 zeroes of the polynomial. Relation between the Zeroes and the coefficients of quadratic polynomial: Let us say that α and β are any two zeroes of the quadratic polynomial p(x) = ax^2 + bx + c , where a <> 0 Now we can say that x - α and x – β are the factors of the polynomial p(x) So we say that ax^2 + bx + c = k . (x – α). ( x – β ), where k is any constant. ax^2 + bx + c = k .x^2 - k. (α + β). X + k .α .β so k = a, -k. (α + β) = b and k. α .β = c
Learn More :- Divergence Theorem
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putting the value of k = a, we get (α + β) = -b/a and α .β = c/a So we say that sum of zeroes = -b/ a = - ( coefficient of x) / ( coefficient of x ^2) Similarly product of zeroes = c/a = constant term / ( coefficient of x ^2) Example 1: Subtract the following two polynomials 7x3+4x2+3x+1 and 2x3+2x2+x+1 Solution: Given: 7x3+4x2+3x+1 and 2x3+2x2+x+1 Solving subtraction of two polynomials = 7x3+4x2+3x+1 - (2x3+2x2+x+1) Step 1: Find the opposite polynomial of the subtracted term Opposite of (2x3+2x2+x+1) subtracted polynomial = -2x3-2x2- x - 1 Step 2: Add the polynomials Addition = (7x3+4x2+3x+1) - (2x3+2x2+x+1) = 7x3+4x2+3x+1 -2x3-2x2 - x -1 (+) -----------------------5x3 +2x2+2x+0 ------------------------Add the equal exponential variables = 5x3+2x2+2x+0 = 5x3+2x2+2x
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