NCERT Solutions Class 9 Maths Exercise 2.1 — Introduction to Linear Polynomials

NCERT Class 9 Mathematics — Ganita Manjari Part I, Introduction to Linear Polynomials, Exercise Set 2.1. All 5 questions solved.

This first exercise is vocabulary, and it is worth getting exactly right, because the rest of the chapter — and the whole of the next one — leans on it. Three definitions carry all five questions:

• A polynomial in one variable is built from a variable and whole-number powers of it, combined with numbers, e.g. $5y^3 + y^2 + 2y – 1$.
• Its degree is the highest power of the variable that actually appears.
• The coefficient of a particular power is the number multiplying it, and the constant term is the number with no variable attached — the coefficient of $x^0$.

The book names polynomials by degree: degree $1$ is linear, degree $2$ quadratic, degree $3$ cubic, and a lone number is a constant polynomial of degree $0$.

Key insight. A coefficient is attached to a power, not to a position. Read $4z^3 + 5z^2 – 11$ as $4z^3 + 5z^2 + 0 \cdot z – 11$ and the missing terms become visible. Almost every mistake in this exercise comes from answering about the term that happens to be written there rather than about the power that was asked for.

Question 1

Find the degrees of the following polynomials:

(i) $2x^2 – 5x + 3$    (ii) $y^3 + 2y – 1$    (iii) $-9$    (iv) $4z – 3$

Solution. The degree is the highest power of the variable present. Scan each expression for the largest exponent — not the largest coefficient, and not the number of terms.

(i) The powers present are $2$ and $1$ (and $0$ for the constant $3$). The highest is $2$, so this is a quadratic polynomial.

(ii) The powers present are $3$ and $1$. The highest is $3$, so this is cubic. Note that there is no $y^2$ term; a missing power does not affect the degree, which only cares about the highest one that is there.

(iii) $-9$ has no variable written at all. It can be written as $-9x^0$, and since $x^0 = 1$ for any non-zero $x$, this is a polynomial of degree $0$ — a constant polynomial.

(iv) The highest power of $z$ is $1$, so this is a linear polynomial. Linear polynomials are what the rest of the chapter is about.

Question 2

Write polynomials of degrees $1$, $2$ and $3$.

Solution. Any polynomial works provided the highest power present is the one asked for, and that the coefficient of that highest power is not zero — otherwise the term vanishes and the degree drops.

Degree 1 (linear): the variable appears only to the first power.

$$3x + 7$$

Degree 2 (quadratic): an $x^2$ term must be present; the lower terms are optional.

$$x^2 + 5x + 1$$

Degree 3 (cubic): a $y^3$ term must be present.

$$5y^3 + y^2 + 2y – 1$$

Since the question says “polynomials” rather than “linear polynomials”, you may use any letter as the variable, and you may leave out any of the lower powers: $4z^3 – 2$ is perfectly good as a cubic.

Question 3

What are the coefficients of $x^2$ and $x^3$ in the polynomial $x^4 – 3x^3 + 6x^2 – 2x + 7$?

Solution. Find the term containing each power and read off the number multiplying it, with its sign.

The term in $x^2$ is $+6x^2$, so its coefficient is $6$.

The term in $x^3$ is $-3x^3$. The minus sign belongs to the coefficient: the polynomial is $x^4 + (-3)x^3 + 6x^2 + (-2)x + 7$, so the coefficient of $x^3$ is $-3$.

Question 4

What is the coefficient of $z$ in the polynomial $4z^3 + 5z^2 – 11$?

Solution. The question asks for the coefficient of $z$ — that is, of $z^1$. Go looking for a $z$ term:

$$4z^3 \;+\; 5z^2 \;-\; 11$$

There is no term in $z$ at all. The polynomial jumps straight from $z^2$ to the constant.

A missing term is not an absent question, though. Writing every power out in full,

$$4z^3 + 5z^2 + 0 \cdot z – 11$$

which is the same polynomial, since $0 \cdot z = 0$. So the coefficient of $z$ is $0$.

Question 5

What is the constant term of the polynomial $9x^3 + 5x^2 – 8x – 10$?

Solution. The constant term is the one with no variable attached — the part of the polynomial that does not change when $x$ changes. Here that is $-10$.

The sign matters. The polynomial is $9x^3 + 5x^2 + (-8)x + (-10)$, so the constant term is $-10$, not $10$.

A useful check: the constant term is the value the polynomial takes at $x = 0$, because every other term contains a factor of $x$ and vanishes.

$$9(0)^3 + 5(0)^2 – 8(0) – 10 = -10 \; \checkmark$$

Common mistakes

• Question 1(iii), calling $-9$ “not a polynomial”. A single number is a polynomial — a constant polynomial, of degree $0$, because it can be written as $-9x^0$. (The one exception is the polynomial $0$ itself, whose degree is left undefined.)
• Question 1, counting terms instead of powers. $2x^2 – 5x + 3$ has three terms but degree $2$. The degree is the highest exponent, and the two numbers are unrelated.
• Question 3, dropping the sign. Answering $3$ for the coefficient of $x^3$. The sign in front of a term belongs to its coefficient, so it is $-3$; getting this wrong reverses the behaviour of the whole term.
• Question 3, answering about $x^4$. The first term is the one the eye lands on, but the question named $x^2$ and $x^3$. Locate the power first, then read the number.
• Question 4, answering $5$ or $4$. These are the coefficients of $z^2$ and $z^3$. The question asked about $z$, which is $z^1$, and there is no such term — so the answer is $0$. Writing the polynomial out with every power shown makes the gap obvious.
• Question 5, answering $10$. The polynomial ends $\ldots – 10$, so the constant term is $-10$. Substituting $x = 0$ is a quick and reliable check.

Practise next

• Exercise Set 2.2 — puts these definitions to work by evaluating polynomials at given values of the variable, and then using linear polynomials to solve word problems.
• The End-of-Chapter Exercises, Questions 1 and 2 — Question 1 asks you to build a polynomial to a specification of degree and coefficient, which is this exercise run backwards, and Question 2 evaluates polynomials as Exercise Set 2.2 does.