Chapter 2 — Polynomials

What is a polynomial? A polynomial in one variable x is an expression of the form a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where the coefficients a_i are real numbers (or from some field) and n is a non‑negative integer. Each a_i x^i is called a term. The highest power n with a_n ≠ 0 is the degree of the polynomial.

Parts of a polynomial: coefficient (a_i), variable (x), exponent (i), constant term (a_0).

Classification by degree: degree 0 — constant polynomial (e.g., 5); degree 1 — linear (e.g., 3x − 2); degree 2 — quadratic (e.g., x^2 + x + 1); degree 3 — cubic, etc. A polynomial with one term is a monomial, two terms is a binomial, three terms is a trinomial.

Zero polynomial: the polynomial 0 (all coefficients zero) is called the zero polynomial; its degree is not defined (or sometimes said to be −∞).

Standard form and like terms: Write terms in descending powers of x to get the standard form. Combine like terms (same power of x) by adding coefficients.

Evaluation and zeros: To evaluate p(x) at x = a, substitute a for x and compute p(a). If p(a) = 0, then a is a root (zero) of the polynomial.

Important idea (Remainder Theorem): When a polynomial p(x) is divided by (x − a), the remainder is p(a). This links evaluation to division.

In Class 9 Mathematics, a polynomial in one variable x is an expression of the form a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where a_i are real numbers and a_n ≠ 0. The highest power n with nonzero coefficient is the degree of the polynomial.

Polynomials are classified in two common ways:

• By degree (highest power of x):
  • Constant polynomial: degree 0 (e.g., 5)
  • Linear polynomial: degree 1 (e.g., 2x + 3)
  • Quadratic polynomial: degree 2 (e.g., x^2 + 4x + 4)
  • Cubic polynomial: degree 3 (e.g., x^3 − 2x^2 + x)
  • Quartic: degree 4; and so on (degree n gives an n-th degree polynomial).
  • Zero polynomial (0): special case — its degree is not defined in CBSE convention.
• By number of nonzero terms:
  • Monomial: one term (e.g., 7x^3)
  • Binomial: two terms (e.g., x + 5)
  • Trinomial: three terms (e.g., x^2 + 2x + 1)
  • Polynomial (general): more than three terms (e.g., x^4 − x^3 + 3x − 2)

Other useful concepts:

• Leading coefficient: the coefficient a_n of the highest-degree term. If a_n = 1 the polynomial is called monic.
• Standard form: arrange terms in descending powers of x: a_n x^n + ... + a_1 x + a_0.
• Operations and degree rules: degree(p + q) ≤ max(degree(p), degree(q)); degree(p · q) = degree(p) + degree(q) provided neither is the zero polynomial.

This classification helps in solving, graphing and modeling problems. For example, many physical relationships are modeled by linear or quadratic polynomials (straight-line motion, area formulas, projectile motion approximations, revenue/cost functions, etc.).

What is a polynomial? A polynomial in one variable x is an expression of the form a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 where each a_i is a constant (real number) and n is a non‑negative integer. Each a_i x^i is called a term (or monomial), and a_i is its coefficient.

Coefficient: The coefficient of a term is the constant multiplying the variable power. Example: in 5x^3 − 2x + 4, the coefficient of x^3 is 5, of x is −2, and of x^0 (the constant term) is 4. If a power of x is missing, its coefficient is 0 (e.g., in x^3 + 2, the coefficient of x^2 is 0).

Degree of a term (monomial): For a single term a x^k (a ≠ 0) the degree is k (the exponent of x).

Degree of a polynomial (single variable): After combining like terms, the degree of a nonzero polynomial is the highest exponent of x that appears with a nonzero coefficient. For f(x) = a_n x^n + ... + a_0 with a_n ≠ 0, degree(f) = n, and a_n is called the leading coefficient. A nonzero constant has degree 0. The zero polynomial (0) has no well‑defined degree (often said to be "undefined" or −∞).

Multivariable polynomials: For a monomial x^a y^b z^c, the degree is a + b + c (sum of exponents). The degree of a multivariable polynomial is the maximum degree among its monomials.

Key properties:

• To determine degree, first simplify/combine like terms.
• For two nonzero polynomials f and g: degree(f + g) ≤ max(degree(f), degree(g)). Equality may fail if leading terms cancel.
• For nonzero polynomials: degree(f · g) = degree(f) + degree(g). The leading coefficient of the product is the product of leading coefficients.
• Leading term (a_n x^n) determines end behavior: for large |x| the polynomial behaves like its leading term.

Practical note: Always write the polynomial in descending powers (highest degree first) to identify the degree and leading coefficient quickly.

Definition: A zero (or root) of a polynomial f(x) is a number 'a' such that f(a) = 0. Graphically, zeros are the x-coordinates where the graph of y = f(x) meets the x-axis.

Why zeros matter: Zeros help us factor polynomials, solve equations, find intercepts in graphs and model real situations (like times when profit is zero or when a projectile hits the ground).

Key ideas and methods:

• Remainder Theorem: If a polynomial f(x) is divided by (x − a), the remainder is f(a). So f(a) = 0 implies (x − a) divides f(x).
• Factor Theorem: (x − a) is a factor of f(x) if and only if f(a) = 0. This is used to find factors and reduce degree step by step.
• Multiplicity: If (x − a)^k is a factor of f(x), then a is a root of multiplicity k. If k is odd the graph crosses the x-axis at a; if k is even it touches and turns around at a.
• Number of roots: A non-zero polynomial of degree n has at most n real roots. (Over the complex numbers it has exactly n roots counting multiplicity.)

How to find zeros (practical steps):

1. Try simple integer candidates using the Remainder Theorem (factor pairs of the constant term are good starting points).
2. When you find a zero a, divide the polynomial by (x − a) (long division or synthetic division) to lower the degree.
3. Factor the resulting polynomial further (quadratic formula, factorisation, or repeated application of the factor theorem) to find all zeros.

Connections with coefficients (quadratic): For ax^2 + bx + c with roots r1 and r2: r1 + r2 = −b/a and r1·r2 = c/a. These relations help check answers or form equations from given roots.

What is a polynomial: A polynomial in x is an expression of the form a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 where coefficients a_i are real numbers and n is a nonnegative integer. The highest power with nonzero coefficient is the degree.

Basic operations

• Addition and subtraction: Combine like terms (terms with the same power of x) by adding or subtracting their coefficients. Degree of the result is at most the maximum degree of the operands.
• Multiplication: Multiply every term of the first polynomial by every term of the second and then combine like terms. Exponents add when you multiply powers of x: x^m * x^n = x^{m+n}. Degree of product = sum of degrees.
• Division (Division algorithm): For polynomials f(x) and g(x) with g(x) not zero, there exist unique q(x) and r(x) such that f(x) = g(x)q(x) + r(x) where either r(x)=0 or deg r < deg g. You can perform polynomial long division or synthetic division when dividing by linear terms of the form x - a.
• Remainder theorem: When a polynomial f(x) is divided by (x - a), the remainder is f(a). If f(a)=0, then (x - a) is a factor of f(x) (factor theorem).

Quick rules and consequences

• Degrees: deg(sum) ≤ max(deg(f),deg(g)); deg(product) = deg(f)+deg(g) except when leading coefficients cancel.
• Zero polynomial has undefined degree (sometimes called -infinity) and all coefficients zero.

These operations let you simplify polynomial expressions, factor polynomials, solve polynomial equations and understand how polynomial expressions model quantities in real life.

Statement: For any two polynomials p(x) and g(x) (g(x) ≠ 0), there exist unique polynomials q(x) (quotient) and r(x) (remainder) such that

p(x) = g(x)·q(x) + r(x), where either r(x) = 0 or deg r(x) < deg g(x).

What it means (intuitively): When you divide one polynomial by another you get a quotient polynomial and a remainder polynomial just like with integers. The remainder’s degree is always smaller than the divisor’s degree.

Division procedure (long division / repeated subtraction):

1. Arrange both polynomials in descending powers of x (fill missing terms with 0 coefficients).
2. Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient.
3. Multiply the divisor by that term and subtract the result from the dividend.
4. Bring down the next term(s) and repeat until the degree of the remaining polynomial (the current remainder) is less than the degree of the divisor.
5. The final leftover is r(x); the collected terms are q(x).

Proof idea of existence and uniqueness (sketch): Existence is shown by performing the long division algorithm—each step reduces the degree of the dividend or stops when remainder degree is smaller than divisor. Uniqueness: if p = gq1 + r1 = gq2 + r2 then g(q1 − q2) = (r2 − r1). The left side is divisible by g, but the degree of (r2 − r1) is less than deg g unless r1 = r2, forcing q1 = q2 and r1 = r2.

Important corollaries:

• Remainder Theorem: The remainder on dividing p(x) by (x − a) is p(a).
• Factor Theorem: (x − a) is a factor of p(x) iff p(a) = 0.

Tips: For division by linear factors (x − a), use synthetic division (fast). For higher-degree divisors, use polynomial long division. Keep coefficients organized (use zeros for missing powers).

Definition: The Remainder Theorem gives a quick way to find the remainder when a polynomial f(x) is divided by a linear polynomial of the form (x − a).

Statement: If a polynomial f(x) is divided by (x − a), then the remainder is f(a).

Why it works (short proof): By the division algorithm for polynomials, f(x) can be written as

f(x) = (x − a)·q(x) + r,

where q(x) is the quotient and r is the remainder. Because the divisor (x − a) has degree 1, r is a constant. Substituting x = a gives f(a) = r. Hence the remainder equals f(a).

Corollary (Factor Theorem): (x − a) is a factor of f(x) if and only if f(a) = 0. That is, f(a) = 0 ⇔ (x − a) divides f(x) exactly.

Generalizations:

• If the divisor is of the form (ax − b), then the remainder is f(b/a) (substitute x = b/a).
• For a divisor g(x) of degree m, division gives f(x) = g(x)·q(x) + r(x) where deg r(x) < m.

How to use the theorem (steps):

• Identify the linear divisor (x − a). If divisor is (ax − b), rewrite as x − (b/a) and use a = b/a.
• Compute f(a) by substituting x = a in the polynomial.
• That value f(a) is the remainder.

Practical notes for Class 9 students: The theorem avoids long polynomial division for linear divisors and is often used to check whether a given linear expression is a factor of a polynomial (use the Factor Theorem).

Definition (Factor Theorem): For a polynomial p(x), (x − a) is a factor of p(x) if and only if p(a) = 0.

Relation to Remainder Theorem: When p(x) is divided by (x − a), the remainder is p(a). So p(a) = 0 implies remainder 0, hence (x − a) divides p(x).

Proof sketch: By polynomial division, p(x) = (x − a)q(x) + r where q(x) is a polynomial and r is a constant. Put x = a to get p(a) = r. Thus p(a) = 0 ⇔ r = 0 ⇔ (x − a) is a factor.

How to use it (step-by-step):

• Given p(x), list possible rational roots (for integer-coefficient polynomials these are divisors of the constant term divided by divisors of the leading coefficient).
• Evaluate p(a) for each candidate a. If p(a) = 0, then (x − a) is a factor.
• Divide p(x) by (x − a) (long division or synthetic division) to obtain a lower-degree polynomial q(x).
• Repeat the process on q(x) until fully factored.

Notes for Class 9: Often you will try integer candidates that divide the constant term (±1, ±2, ±3, ...). For quadratics, finding one root lets you factor it into two linear factors.

Example illustration (quick):

p(x) = x^2 − 5x + 6. Try x = 2: p(2)=4−10+6=0 ⇒ (x−2) is a factor.
Divide by (x−2) ⇒ quotient (x−3) ⇒ p(x)=(x−2)(x−3).

What is factorization? Factorization of a polynomial means writing it as a product of two or more polynomials of lower degree. For example, x^2 + 5x + 6 = (x + 2)(x + 3).

Why it matters: Factoring helps find zeros (roots) of polynomials, simplifies algebraic expressions, and solves equations and real-life problems (areas, arrangements, simplification).

Key idea — zeros and factors: If p(a) = 0 for a polynomial p(x), then (x − a) is a factor of p(x). Conversely, if (x − a) is a factor, then a is a zero of p(x).

Common methods used in Class 9:

1. Taking out the common factor: Factor out the greatest common divisor of all terms. Example: 4x^3 − 8x^2 + 12x = 4x(x^2 − 2x + 3).
2. Factorization of quadratic trinomials (ax^2 + bx + c): Use the split-the-middle-term method: find two numbers whose product = a·c and sum = b, then split the middle term and factor by grouping. Example: 6x^2 + 11x + 3: numbers 9 and 2 because 9·2 = 18 (=6·3) and 9 + 2 = 11. So 6x^2 + 9x + 2x + 3 = 3x(2x + 3) + 1(2x + 3) = (2x + 3)(3x + 1).
3. Difference of squares: a^2 − b^2 = (a − b)(a + b). Example: x^2 − 9 = (x − 3)(x + 3).
4. Perfect square trinomials: a^2 + 2ab + b^2 = (a + b)^2 and a^2 − 2ab + b^2 = (a − b)^2. Example: x^2 + 6x + 9 = (x + 3)^2.
5. Factorization by grouping: Used when a polynomial has four terms. Group terms in pairs and factor each pair. Example: x^3 + 3x^2 + x + 3 = x^2(x + 3) + 1(x + 3) = (x + 3)(x^2 + 1).

Practical steps to factor a quadratic ax^2 + bx + c:

1. Compute product a·c.
2. Find two integers m and n such that m·n = a·c and m + n = b.
3. Write bx as mx + nx and factor by grouping.

Tip about discriminant (useful idea): For ax^2 + bx + c, the quantity D = b^2 − 4ac tells whether real linear factors exist: if D > 0 two distinct real linear factors; D = 0 one repeated factor; D < 0 no real linear factors (irreducible over reals).

Consider a quadratic polynomial f(x) = ax² + bx + c with a ≠ 0. Let its zeros (roots) be α and β. If we factor the quadratic using its zeros we get f(x) = a(x − α)(x − β).

Expanding: a(x − α)(x − β) = ax² − a(α + β)x + aαβ. Comparing coefficients with ax² + bx + c gives the fundamental relations:

α + β = −b/a, and αβ = c/a.

These relations are powerful: they let you form a quadratic from given zeros (a(x − α)(x − β)), check roots quickly, and connect geometric features of the parabola with algebraic coefficients. For example the axis of symmetry (x-coordinate of vertex) is x = (α + β)/2 = −b/(2a). The discriminant D = b² − 4ac determines the nature of roots: D > 0 → two distinct real roots, D = 0 → one (repeated) real root, D < 0 → two complex conjugate roots.

Another useful derived relation for the distance between roots is |α − β| = sqrt((α + β)² − 4αβ) = sqrt(b² − 4ac)/|a|. This ties the spread of the zeros to the discriminant.

In Class 9 Polynomials, the section 'Applications and Problems' shows how polynomial expressions model real situations and how algebraic tools (factorisation, division algorithm, remainder theorem and factor theorem) are used to solve practical questions. Typical tasks are: form a polynomial from a word description, find zeroes, check or find factors, determine remainders on division, compare coefficients to find unknowns, and use polynomial expressions to represent areas, revenues or other quantities.

Key methods:

• Model: Express the quantity (length, area, profit, etc.) as a polynomial in a variable x.
• Simplify or expand the polynomial to standard form.
• Use factorisation or the factor theorem to find roots/zeros or simplify expressions.
• Apply the division algorithm/ remainder theorem to find remainders when divided by linear factors.
• Compare coefficients or substitute specific x-values to determine unknown constants.

Practical problem types include geometric problems (areas/perimeters expressed in x), algebraic verification problems (proving identities), constructing polynomials with given zeros, and using remainders/factors to test divisibility. Strategies: always try substitution (for remainder/factor checks), factor where possible, and interpret roots in context (e.g., a root may represent a dimension giving zero area — often not physically admissible but useful algebraically).