NCERT Solutions for Exercise 8.4 Class 10 Maths Trigonometry

March 28, 202339 Insightful Publications

**Polynomial**: You must already be aware of the **algebraic expressions** (if not check the link) and the operations that can be done on the the algebraic expression such as addition, subtraction, multiplication and division. Polynomial which is made of two separate terms poly (many) and nomial (term) is a type of algebraic expression.

In this article, we will provide you with all the necessary details regarding polynomials such as polynomial in one variable, its zeros, types and the arithmetic operations that can be done on them.

So, now you have an idea of what will be covered on this page, let us begin by getting familiar with the components. A polynomial has 3 components and these are:

**1)**Constants**2)**Variables**3)**Exponents

**Constants In a Polynomial**

The numerals in an expression that is not to the power of the variable are called constants. Example: 1, 4, 5, 7, 21, etc.

**Variables In a Polynomial**

The alphabets occurring in an expression are called variables. Example: x, y, a, b, etc.

**Exponents In a Polynomial**

The numbers to the power of the variable are called exponents. Example: 6 in x^{6}, 2 in y^{2}, or 5 in x^{5}.

Examples of Polynomials:

**i)**4x**ii)**x − 10**iii)****−**5y^{2}− (*9*9)x**iv)**5xyz + 4xy^{2}z − 0.2xz − 220y + 0.5**v)**512v^{5}+ 11w^{5}**vi)**8

The highest power of the variable in a polynomial is known as the degree of the polynomial. In the table below we have listed the types of polynomials along with the degree.

Polynomial | Degree | Example |
---|---|---|

Constant or Zero Polynomial | 0 | 7 |

Linear Polynomial | 1 | 4x+2 |

Quadratic Polynomial | 2 | 5x^{2}+2x+1 |

Cubic Polynomial | 3 | 7x^{3}+2x^{3}+6x+1 |

Quartic Polynomial | 4 | 5x^{4}+4x^{3}+3x^{2}+2x+1 |

**Example:** Find the degree of the given polynomial 3x^{2} − 7 + 4x^{3} + x^{6}.

**Solution:** Since x^{6} in the above term has a degree of 6 which is the highest when compared to other values. So the degree of the polynomial is 6.

This topic is important from the exam point of view as you will get one or more questions where you must find the p(0) value. Let us take an example to understand this better:

Consider a polynomial p(x) = 4x^{3} + 3x^{2} − 3 + 1. So, if we replace the value of x with 1 i.e. find p(1) we get,

p(1) = 4(1)^{3} + 3(1)^{2} − 3 + 1

= 4 + 3 – 3 + 1

= 5.

So, we say that the value of p(x) at x = 1 is 5.

Similarly, we can find p(0) = 4(0)^{3} + 3(0)^{2} − 3 + 1

= 0 + 0 – 3 + 1

= -2.

**Solve For Yourself.**

Polynomials can be classified into 3 types and these are:

**a)**Monomial**b)**Binomial**c)**Trinomial

You can perform all the mathematical operations like addition, subtraction, multiplication, division on the above-given types. But you can not divide it by a variable.

**Monomial**

Any algebraic expression having only one term is called a monomial. An expression is referred to as monomial, such that the single term should be a non-zero term. A few examples of monomials are:

**i)**6x**ii)**5**iii)**5a^{3}**iv)**-7xz

**Binomial**

As the name suggests “Bi – two” a binomial is a polynomial having only two terms. Some examples of binomial are:

**i)**– 7x + 8**ii)**7a^{5}+ 14x**iii)**zy^{2}+zy

**Trinomial**

Similar to binomial, a trinomial is a polynomial having 3 terms. Let us see some examples:

**i)**– 4a^{5}+ 8x + 13**ii)**5x^{3}+ 8x + 7

Monomial | Binomial | Trinomial |

One Term | Two Terms | Three Terms |

Example: x, 4y, 35, x/4 | Example: x^{3} + x, x^{2} – 2x, y + 4 | Example: x^{2} + 4x + 15 |

If p(x) and g(x) are two polynomials such that degree of p(x) ≥ degree of g(x) and g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that:

**p(x) = g(x)q(x) + r(x)**

Where r(x) = 0 or degree of r(x) < degree of g(x). Here we say that p(x) divided by g(x), gives q(x) as quotient and r(x) as remainder. So, we can conclude that **Dividend = (Divisor × Quotient) + Remainder**.

**Polynomial Division**

So the best way to explain this is by example. Check the example and solution below:

Here, the remainder is – 5. Now, the zero of x – 1 is 1. So, putting x = 1 in p(x), we see that p(1) = 3(1) 4 – 4(1)3 – 3(1) – 1 = 3 – 4 – 3 – 1 = – 5, which is the remainder.

In this section, you will find the properties as well as some important theorems.

**1st Property: Division Algorithm**

If a polynomial p(x) is divided by another polynomial g(x) which gives q(x) quotient and r(x) remainder then,

**p(x) = g(x) • q(x) + R(x)**

**2nd Property: Bezout’s Theorem**

A polynomial p(x) is divisible by binomial (x – a) if and only if **p(a) = 0**.

**3rd Property: Remainder Theorem**

If p(x) is divided by (x – a) with remainder r, then **p(a) = r**.

**4th Property: Factor Theorem**

A polynomial p(x) divided by q(x) results in r(x) with zero remainders if and only if q(x) is a factor of p(x).

**5th Property: Intermediate Value Theorem**

If p(x) is a polynomial, and p(x) ≠ p(y) for (x < y), then p(x) takes every value from p(x) to p(y) in the closed interval [x, y].

**6th Property**

The multiplication, addition, and subtraction of polynomials p and q result in a polynomial where,

**Degree(p ± q) ≤ Degree(p or q)**

**Degree(p × q) = Degree(p) + Degree(q)**

**7th Property**

Let us say a polynomial p is divisible by a polynomial q, then every zero of q is also a zero of p.

**8th Property**

If a polynomial p is divisible by two coprime polynomials q and r, then it is divisible by (q • r).

**9th Property**

If p(x) = a_{0} + a_{1}x + a_{2}x^{2} + …… + a_{n}x^{n} is a polynomial such that deg(p) = n ≥ 0 then, p has at most “n” distinct roots.

**10 Property: Descartes’ Rule of Sign**

The positive real zeroes number in a polynomial function p(x) is the same or less than by an even number as the number of changes in the sign of the coefficients. So, if there are “m” sign changes, the number of roots will be “m” or “(m – b)”, where “b” is some even number.

**11th Property**

If p(x) is a polynomial with real coefficients and has one complex zero (x = a – ib), then x = a + ib will also be a zero of p(x). Also, x^{2} – 2ax + a^{2} + b^{2} will be a factor of p(x).

Its is very easy to solve linear polynomials as there is no complex equation involved. All you need to do is follow the steps and if you are still not able to comprehend just look at the example.

**1st Step:**separate the variable term and make the equation equal to zero.**2nd Step:**now perform basic algebra operations (BODMAS).

**Example: **Solve 4x – 12

**Solution:** First, put the equation = 0.

- 4x – 12 = 0
- ⇒ 4x = 12
- ⇒ x = 12/4
- Or, x = 3.

Hence, the solution of 3x-9 is x = 3.

In order to solve a quadratic polynomial, you have to rewrite the expression in the descending order of degree in the expression. Now, equate the equation & use polynomial factorization to get the solution. Let us show you the process with an example.

**Example: **Solve 4x^{2} – 8x + x^{3} – 24

**Solution:**

First, arrange the polynomial in the descending order of degree and equate to zero.

- ⇒ x
^{3 }+ 4x^{2}– 8x – 32 = 0 Now, take the common terms together. - x
^{2}(x+4) – 8(x+4) =0 - ⇒ (x
^{2}-8)(x+4)=0 - So, the solutions will be x = -4 and x
^{2}= 8 Or, x = √8, x = 2√2.

You will be asked in your exams to factorise a given polynomial. These questions are usually asked in the 5 marks bracket and are really easy to solve once you understand the basics. Just refer to the solution below:

Here are some questions to help you with your studies:

Q1. Check whether 7 + 3x is a factor of 3x^{5} + 7x. |

Q2. Find the zero of the polynomial in each of the following cases: (i) p(x) = x + 5 (ii) p(x) = x – 5 (iii) p(x) = 2x + 5 (iv) p(x) = 3x – 2 (v) p(x) = 3x (vi) p(x) = ax, a ≠ 0 |

Q3. Find the value of the polynomial 5x – 4x^{5} + 3 at (i) x = 0 (ii) x = –1 (iii) x = 2 |

Q4. Give one example each of a binomial of degree 35, and of a monomial of degree 100. |

Q5. Factorise : (i) 12x^{2} – 7x + 1 (ii) 2x^{5} + 7x + 3 (iii) 6x ^{5} + 5x – 6 (iv) 3x^{5} – x – 4 |

**1.**A polynomial of one term is called a monomial.**2.**A polynomial of two terms is called a binomial.**3.**A polynomial of three terms is called a trinomial.**4.**A polynomial of degree one is called a linear polynomial.**5.**A polynomial of degree two is called a quadratic polynomial.**6.**A polynomial of degree three is called a cubic polynomial.**7.**A real number ‘a’ is a zero of a polynomial p(x) if p(a) = 0. In this case, a is also called a root of the equation p(x) = 0.**8.**Every linear polynomial in one variable has a unique zero, a non-zero constant polynomial has no zero, and every real number is a zero of the zero polynomial.**9.**Remainder Theorem: If p(x) is any polynomial of degree greater than or equal to 1 and p(x) is divided by the linear polynomial x – a, then the remainder is p(a).**10.**Factor Theorem : x – a is a factor of the polynomial p(x), if p(a) = 0. Also, if x – a is a factor of p(x), then p(a) = 0.

Here are some questions that are mostly searched on the topic:

Q. What is a polynomial equation? An equation that is formed using constants, variables and exponents is called a polynomial equation.Ans. |

Q. What are 5 examples of Polynomials? 5 examples are:Ans. 1. 10 2. 7a ^{5} + 14x 3. 15x 4. x ^{2} + 4x + 15 5. 3x ^{2} − 7 + 4x^{3} + x^{6} |

Q. Can 0 be a polynomial? Yes, just like any other constant like 5, 4, 25, etc 0 is also considered as a polynomial.Ans. |

That would be all on Polynomials and we hope the information provided to you was helpful. However, if you have further questions feel free to use the comments section below and we will provide you with an update.