You also have studied how to factorise some algebraic expressions. In this chapter, we **ncert 10th math 9.1 full** start our study with a particular type of algebraic expression, called polynomial, and the terminology related to it. We shall also study the Remainder Theorem and Factor Theorem and their use in the factorisation of polynomials.

In addition to the above, we shall study some more algebraic identities and their use in factorisation and in evaluating some given expressions. We use the letters x, y, z. Notice that 2x, 3x, x, x 2 are algebraic expressions. All these expressions are of the form a constant x. Now suppose we want to write an expression which is a constant a variable and we do not know what the constant is. In such cases, we write the constant as a, b, c.

So the expression will be ax, say. However, there is a difference between a letter denoting a constant and a letter denoting a variable. The values of the constants remain the same throughout a particular situation, that is, the values of the constants do not change in a given problem, but the value of a variable can keep changing.

Now, consider a square of side 3 units see Fig. What is its perimeter? You know that the perimeter of a square is the sum of the lengths of its four sides. Here, each side is 3 units. So, its perimeter is 4 3, i. What will be the perimeter if each side of the square is 10 units? The perimeter is 4 10, i. In case the **ncert 10th math 9.1 full** of each side is **ncert 10th math 9.1 full** units see Fig.

So, as the length of the side varies, the perimeter varies. Can you find the area of the square PQRS? Note that, all the algebraic expressions we have considered so far have only whole numbers as the exponents of the variable.

Expressions of this form are called **ncert 10th math 9.1 full** in one variable. In the examples above, the variable is x. This polynomial has 4 terms, namely, x3, 4x2, 7x and 2. Each term of a polynomial has a coefficient. It is 1. In fact, 2, 5, 7. The **ncert 10th math 9.1 full** polynomial 0 is called the zero polynomial. This plays a very important role in the collection of all polynomials, as you will see in the higher classes.

Here, the exponent of the second term, i. So, this algebraic expression is not a polynomial. So, is x 3 a polynomial? No, it is not. It **ncert 10th math 9.1 full** also not a polynomial Why?

If the variable in a polynomial is x, we may denote the polynomial by p xor q xor r x. Consider the polynomials 2x, 2, 5x3, 5x2, y and u4. Do you see that each of these polynomials has only one term? Polynomials having only one term are called monomials mono means one. How many terms are there in each of these? Each of these polynomials has only two terms. Polynomials **ncert 10th math 9.1 full** only two terms are called binomials bi means two.

Similarly, polynomials having only three terms are called trinomials tri means. What is the term with the highest power of x? It is 3x7. The exponent of x in this term is 7. We call the highest power of the variable in a polynomial as the degree of the polynomial. The degree of a non-zero constant polynomial is zero. Solution : i The highest power of the variable is 5.

So, the degree of the polynomial is 5. So, the degree of the polynomial is 8. So the exponent of x is 0. Therefore, the degree of the **ncert 10th math 9.1 full** is 0.

Do you see anything common among all of them? The degree of each of these polynomials is one. A polynomial of degree one is called a linear polynomial. **Ncert 10th math 9.1 full,** try and find a linear polynomial in x with 3 terms?

You would not be able to find it because a linear polynomial in x can have at most two terms. Do you agree that they are all of degree two? A polynomial of degree two is called a quadratic polynomial. Can you write a quadratic polynomial in one variable with four different terms? You will find that a quadratic polynomial in one variable will have at most 3 terms. We call a polynomial of degree three a cubic polynomial. How many terms do you think a cubic polynomial in one variable can have?

It can have at most 4 terms. Now, that you have seen what a polynomial of degree 1, degree 2, or degree 3 looks like, can you write down a polynomial in one variable of degree n for any natural number n? What is the degree of the zero polynomial? The degree of the zero polynomial is not defined. So far we have dealt with polynomials in one variable.

We can also have polynomials in more than one variable. You will be studying such polynomials in detail later. Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. Can you find p 1?

What is p 1? You must have observed that the zero of the polynomial x 1 is obtained by equating it to 0, **ncert 10th math 9.1 full.** Now, consider the constant polynomial 5. Can you tell what its zero is? It has no zero because replacing x by any number in 5x0 still gives us 5.

In fact, a non-zero constant polynomial has no zero. What about the zeroes of the zero polynomial? **Ncert 10th math 9.1 full** convention, every real number is a zero of the zero polynomial.

Example 4 may have given you some idea. Example 5 : Verify whether 2 and 0 are zeroes of the polynomial x2 2x. Solution : Let. Hence, 2 and 0 are both zeroes of the polynomial x2 2x. Let us now list our observations: i A zero of a polynomial need not be 0.

Verify whether the following are zeroes of the polynomial, indicated against. You know that when we divide 15 by 6, we get the quotient 2 and remainder 3. Do you remember how this fact is expressed? Here the remainder is 0, and we say that 6 is a factor of 12 or 12 is a multiple of 6. Now, the question is: can we divide one polynomial by another?

To start with, let us try and do this when the divisor is a monomial. We. We see that we cannot divide 1 by x to get a polynomial term. So in this case we stop here, and note that 1 is the remainder. Since the remainder is not zero, it is not a factor. Now let us consider an example to see how we can divide a polynomial by any non-zero polynomial.

Real Numbers Class 10 has total of four exercises consists of 18 Problems. Hence the distance between the two ships is m. Hence the length of the string is m. Good app and site. I can understand the problem which was very difficult.

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