11TH CLASS MATHS CH#4 #EX4.5 #q1 -- 11 REMAINDER & FACTOR THEOREM #quadraticequation #lecture (64)

The paragraph introduces the concept of polynomials, specifically focusing on first-degree polynomials. The lecturer explains the terms 'variable' and 'constant' in the context of polynomials, noting that variables can take different values for each term, while constants maintain a fixed value. The lecture also touches on operations such as addition, subtraction, and division, and how they apply to polynomials. An example is given to illustrate the definition of a polynomial in the form of a single variable expression, emphasizing that if the highest power term equals zero, the polynomial's degree is not defined and it cannot be negative or fractional. The concept of the degree of a polynomial is also introduced.

This section delves into the Division Algorithm, explaining how to divide any polynomial by a linear polynomial and the outcome of such division, which includes a quotient and a remainder. The lecturer emphasizes that the process involves actual division, but in this case, they are not performing the division to find the remainder. Instead, they extract a value of 'x' and substitute it into the function to determine the remainder. The Factor Theorem is also introduced, which states that if the remainder is zero after division, then 'x - a' is a factor of the polynomial.

The paragraph applies the Remainder and Factor Theorems to specific polynomials. The lecturer demonstrates how to find the remainder when a given polynomial is divided by 'x+1' and 'x-2'. By substituting the values into the polynomial and simplifying, the remainders are calculated. The results are then used to determine whether certain values are factors of the polynomials, with examples showing both when a value is a factor and when it is not.

This section continues with more examples of applying the Remainder and Factor Theorems. The lecturer guides through the process of finding remainders for different polynomials when divided by 'x-1', 'x-2', and 'x+2'. Each example is worked through step by step, showing how to substitute the values and calculate the remainders. The results are then interpreted to determine if the divisors are factors of the polynomials.

The focus of this paragraph is on polynomial division and calculating remainders with specific examples. The lecturer explains the process of dividing polynomials and finding remainders when divided by 'x-a'. The examples include simplifying the polynomials and substituting values to find the remainders. The paragraph also addresses the implications of the remainder being zero, which indicates that 'x-a' is a factor of the polynomial.

The final part of the script wraps up the lecture with a conclusion and some final calculations. The lecturer summarizes the key points covered in the lecture, including the concepts of polynomials, degrees, remainder and factor theorems, and their applications. A final question is posed regarding the value of 'k' when a specific polynomial is divided by 'x-2', and the remainder is given as 14. The solution is worked out, leading to the conclusion of the video lecture.