📈 Introduction to
Linear Polynomial

In everyday life, we constantly deal with expressions that contain variables. When a shopkeeper says “the cost of x notebooks at ₹40 each is 40x,” he is unknowingly using a polynomial! Polynomials are one of the most fundamental building blocks of algebra and appear in nearly every branch of mathematics.

The word polynomial comes from the Greek words poly (many) and nomial (terms). So a polynomial literally means “many terms.” However, even a single term like 5x or a constant like 7 is also considered a polynomial.

An algebraic expression is a combination of constants, variables, and arithmetic operations (+, −, ×). For example: 3x + 5, 2x² − 7x + 1, or just the number 4.

A polynomial is a special type of algebraic expression where:

• The variable has only whole number (non-negative integer) exponents: 0, 1, 2, 3, …
• The coefficients are real numbers.
• There are a finite number of terms.

A linear polynomial is a polynomial of degree 1. It has the general form:

The word “linear” comes from “line” — the graph of every linear polynomial is a straight line. This is why linear polynomials are so important: they model relationships where one quantity changes at a constant rate with respect to another.

A zero (or root) of a polynomial p(x) is a value of x for which p(x) = 0. In other words, it is the value that makes the polynomial equal to zero.

For p(x) = ax + b (where a ≠ 0), set p(x) = 0:

ax + b = 0  →  ax = −b  →  x = −b/a

The Remainder Theorem provides a quick way to find the remainder when a polynomial is divided by a linear polynomial, without actually performing the long division!

When we divide a polynomial p(x) by (x − a), we get:

p(x) = (x − a) × q(x) + r

where q(x) is the quotient and r is the remainder (a constant). Now substitute x = a:

p(a) = (a − a) × q(a) + r = 0 × q(a) + r = r

So p(a) = r (the remainder). That’s it!

The Factor Theorem is a direct consequence of the Remainder Theorem. It gives us a powerful way to check if a linear expression is a factor of a polynomial.

In simple words: if substituting ‘a’ into the polynomial gives zero, then (x − a) divides the polynomial exactly (with no remainder).

Test your understanding! Type any algebraic expression below and the system will tell you if it is a polynomial, its degree, type, and the number of terms.

Enter a linear polynomial to find its zero, or enter any polynomial and a value to find the remainder using the Remainder Theorem!

See how changing the values of a (slope) and b (y-intercept) affects the graph of p(x) = ax + b. The red dot shows the zero of the polynomial!

Click on an option to check your answer. The correct option will turn green.

• Q1. Which of the following is a polynomial?
  • a) x + 1/x
  • b) x² + 3x + 2
  • c) √x + 5
  • d) 1/(x + 1)
  ✅ (b) — All exponents of x are whole numbers (2, 1, 0). The others have negative or fractional exponents.
• Q2. The degree of the polynomial 4x³ − 7x + 9 is:
  • a) 1
  • b) 2
  • c) 3
  • d) 0
  ✅ (c) — The highest power of x is 3 (from the term 4x³).
• Q3. The zero of p(x) = 5x − 10 is:
  • a) −2
  • b) 2
  • c) 10
  • d) 5
  ✅ (b) — 5x − 10 = 0 gives x = 10/5 = 2.
• Q4. A polynomial with two terms is called a:
  • a) Monomial
  • b) Binomial
  • c) Trinomial
  • d) Linear
  ✅ (b) — Bi = 2, so binomial = two terms.
• Q5. The degree of a non-zero constant polynomial is:
  • a) 0
  • b) 1
  • c) Not defined
  • d) −1
  ✅ (a) — A constant like 7 = 7x° has degree 0.
• Q6. The remainder when p(x) = x² + 3x + 2 is divided by (x − 1) is:
  • a) 0
  • b) 2
  • c) 6
  • d) 3
  ✅ (c) — p(1) = 1 + 3 + 2 = 6.
• Q7. If (x − 1) is a factor of p(x), then:
  • a) p(1) = 0
  • b) p(−1) = 0
  • c) p(0) = 1
  • d) p(1) = 1
  ✅ (a) — By the Factor Theorem, (x − a) is a factor means p(a) = 0. Here a = 1.
• Q8. How many zeros can a linear polynomial have?
  • a) 0
  • b) 1
  • c) 2
  • d) Infinite
  ✅ (b) — A linear polynomial (degree 1) has exactly one zero.
• Q9. p(x) = −3x + 9 is a:
  • a) Linear polynomial
  • b) Quadratic polynomial
  • c) Constant polynomial
  • d) Not a polynomial
  ✅ (a) — The highest power of x is 1, so it is linear.
• Q10. The zero of p(x) = −3x + 9 is:
  • a) −3
  • b) 9
  • c) 3
  • d) −9
  ✅ (c) — −3x + 9 = 0 gives x = 9/3 = 3.
• Q11. The coefficient of x² in 5x³ − 2x² + x − 7 is:
  • a) 5
  • b) −2
  • c) 1
  • d) −7
  ✅ (b) — The term with x² is −2x², so its coefficient is −2.
• Q12. If p(x) = x² − 5x + 6 and p(2) = 0, then (x − 2) is:
  • a) A factor of p(x)
  • b) A multiple of p(x)
  • c) The zero of p(x)
  • d) None of these
  ✅ (a) — By the Factor Theorem, p(2) = 0 means (x − 2) is a factor.
• Q13. The zero polynomial is:
  • a) A constant polynomial of degree 0
  • b) A linear polynomial
  • c) A polynomial whose degree is not defined
  • d) Not a polynomial
  ✅ (c) — The zero polynomial (p(x) = 0) has no defined degree.
• Q14. The remainder when p(x) = x² − 5x + 6 is divided by (x − 3) is:
  • a) 0
  • b) 1
  • c) 2
  • d) 6
  ✅ (a) — p(3) = 9 − 15 + 6 = 0. So (x − 3) is a factor.
• Q15. The graph of a linear polynomial is a:
  • a) Straight line
  • b) Parabola
  • c) Circle
  • d) Curve
  ✅ (a) — The graph of p(x) = ax + b is always a straight line.

• Q1. Define a polynomial in one variable.
  A polynomial in one variable is an algebraic expression of the form p(x) = a_nxⁿ + a_n−1xⁿ⁻¹ + … + a₁x + a₀, where a_n, a_n−1, …, a₀ are real numbers (constants), n is a non-negative integer, and x is the variable. All exponents must be whole numbers.
• Q2. What is the degree of the polynomial 7x&sup4; − 3x³ + 2x − 1?
  The degree is 4, because the highest power of x with a non-zero coefficient is x&sup4; (with coefficient 7).
• Q3. Classify by number of terms: (i) 3x² (ii) x + 5 (iii) x² − x + 1
  (i) 3x² has 1 term → Monomial. (ii) x + 5 has 2 terms → Binomial. (iii) x² − x + 1 has 3 terms → Trinomial.
• Q4. Find the zero of p(x) = 7x − 21.
  Set p(x) = 0: 7x − 21 = 0 → x = 21/7 = 3. Verification: p(3) = 7(3) − 21 = 0 ✅
• Q5. Is p(x) = x² + √x a polynomial? Why or why not?
  No. The term √x = x^1/2 has a fractional exponent (1/2), which is not a whole number. A polynomial must have only whole number exponents.
• Q6. State the Remainder Theorem.
  If a polynomial p(x) is divided by the linear polynomial (x − a), then the remainder is p(a). In other words, we simply substitute x = a into the polynomial to find the remainder.
• Q7. Find the remainder when p(x) = x³ − 2x² + x + 1 is divided by (x + 1).
  x + 1 = x − (−1), so a = −1. Remainder = p(−1) = (−1)³ − 2(−1)² + (−1) + 1 = −1 − 2 − 1 + 1 = −3.
• Q8. Is (x − 1) a factor of p(x) = x³ − 1? Justify.
  p(1) = (1)³ − 1 = 1 − 1 = 0. Since p(1) = 0, by the Factor Theorem, (x − 1) is a factor of x³ − 1.
• Q9. What is the difference between the Remainder Theorem and the Factor Theorem?
  The Remainder Theorem says that when p(x) is divided by (x − a), the remainder is p(a). The Factor Theorem is a special case: if that remainder p(a) = 0, then (x − a) is a factor of p(x). The Factor Theorem adds the condition “remainder equals zero.”
• Q10. Can a constant polynomial have a zero? Explain.
  A non-zero constant polynomial (like p(x) = 5) has no zero because 5 ≠ 0 for any value of x. However, the zero polynomial (p(x) = 0) has every real number as its zero, since 0 = 0 is always true.
• Q11. Write a quadratic polynomial whose zeros are 2 and −3.
  If the zeros are 2 and −3, the factors are (x − 2) and (x + 3). So: p(x) = (x − 2)(x + 3) = x² + x − 6. Verify: p(2) = 4 + 2 − 6 = 0 ✅, p(−3) = 9 − 3 − 6 = 0 ✅
• Q12. If p(x) = ax + b has zero at x = 5, and b = −10, find a.
  Zero at x = 5 means: a(5) + (−10) = 0 → 5a − 10 = 0 → 5a = 10 → a = 2. So p(x) = 2x − 10.