Polynomials – Complete Notes for Competitive Exams

1. Introduction

A polynomial is an algebraic expression made up of one or more terms with variables raised to non-negative integer powers.

General form:$$P(x) = a_n x^n + a_{n-1} x^{n-1} + … + a_1 x + a_0$$P(x)=an​xn+an−1​xn−1+…+a1​x+a0​

where $a_i$ai​ are constants and $n$n is a non-negative integer.

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2. Degree of a Polynomial

• Degree = highest power of the variable in the polynomial.
• Example: $P(x) = 5x^4 + 3x^2 – x + 7$P(x)=5×4+3×2−x+7 → Degree = 4

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3. Types of Polynomials

1. Monomial – single term, e.g., $5x^2$5×2
2. Binomial – two terms, e.g., $x + 3$x+3
3. Trinomial – three terms, e.g., $x^2 + x + 1$x2+x+1
4. Polynomial – more than three terms, e.g., $2x^3 + 3x^2 – x + 5$2×3+3×2−x+5

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4. Key Concepts

(a) Remainder Theorem

If a polynomial $P(x)$P(x) is divided by $(x – a)$(x−a), the remainder = $P(a)$P(a)

(b) Factor Theorem

If $P(a) = 0$P(a)=0, then $(x – a)$(x−a) is a factor of $P(x)$P(x)

(c) Sum and Product of Roots

For quadratic polynomial $ax^2 + bx + c = 0$ax2+bx+c=0:

• Sum of roots = $-b/a$−b/a
• Product of roots = $c/a$c/a

(d) Special Products

• $(x + a)(x + b) = x^2 + (a+b)x + ab$(x+a)(x+b)=x2+(a+b)x+ab
• $(x – a)(x – b) = x^2 – (a+b)x + ab$(x−a)(x−b)=x2−(a+b)x+ab

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5. Important Tips

• Always check the highest degree term for degree
• Factorization saves time in exams
• Use Remainder & Factor Theorem for quick evaluation
• Carefully handle signs when expanding or factoring

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Top 25 Practice Questions – Polynomials

Q1.

Find the degree of $P(x) = 3x^4 + 5x^2 – x + 7$P(x)=3×4+5×2−x+7

Q2.

Check if $x – 2$x−2 is a factor of $P(x) = x^3 – 3x^2 + 4x – 8$P(x)=x3−3×2+4x−8

Q3.

Divide $P(x) = x^3 + 4x^2 + 5x + 2$P(x)=x3+4×2+5x+2 by $x + 1$x+1

Q4.

Find the remainder when $x^3 – 2x^2 + 3x – 4$x3−2×2+3x−4 is divided by $x – 1$x−1

Q5.

Factorize $x^2 + 5x + 6$x2+5x+6

Q6.

Factorize $x^2 – 7x + 12$x2−7x+12

Q7.

Solve $x^2 – 3x – 10 = 0$x2−3x−10=0

Q8.

If $x = 1$x=1 is a root of $P(x) = x^3 – 6x^2 + 11x – 6$P(x)=x3−6×2+11x−6, find another factor

Q9.

Divide $2x^3 + 3x^2 – 5x + 6$2×3+3×2−5x+6 by $x + 2$x+2

Q10.

Find the sum and product of roots of $x^2 – 8x + 15$x2−8x+15

Q11.

If $x^3 – 4x^2 + x + 6 = 0$x3−4×2+x+6=0, find a root by trial

Q12.

Factorize $x^3 + 3x^2 – 4x – 12$x3+3×2−4x−12

Q13.

Check whether $x + 3$x+3 is a factor of $x^3 + 4x^2 + x – 6$x3+4×2+x−6

Q14.

Divide $x^4 – 2x^3 + x^2 + x – 2$x4−2×3+x2+x−2 by $x – 1$x−1

Q15.

Solve $x^2 + x – 6 = 0$x2+x−6=0

Q16.

Find the remainder when $2x^3 – 5x^2 + 4x – 1$2×3−5×2+4x−1 is divided by $x – 2$x−2

Q17.

Factorize $x^2 – 9$x2−9

Q18.

Find the roots of $x^2 – 4x + 4$x2−4x+4

Q19.

Divide $x^3 – 2x^2 – x + 2$x3−2×2−x+2 by $x – 2$x−2

Q20.

Check whether $x – 1$x−1 is a factor of $x^3 – x^2 – x + 1$x3−x2−x+1

Q21.

Factorize $x^3 – 3x^2 – 4x + 12$x3−3×2−4x+12

Q22.

Find the sum of roots of $3x^2 – 5x + 2 = 0$3×2−5x+2=0

Q23.

Factorize $x^2 + 7x + 12$x2+7x+12

Q24.

Divide $x^3 + 6x^2 + 11x + 6$x3+6×2+11x+6 by $x + 1$x+1

Q25.

If $x = -2$x=−2 is a root of $x^3 + x^2 – 4x – 4 = 0$x3+x2−4x−4=0, find other factors

Answer

Answers – Polynomials

Q1. 4
Q2. Yes, $x – 2$x−2 is a factor
Q3. Quotient = $x^2 + 3x + 2$x2+3x+2, Remainder = 0
Q4. Remainder = -2
Q5. $(x + 2)(x + 3)$(x+2)(x+3)
Q6. $(x – 3)(x – 4)$(x−3)(x−4)
Q7. $x = 5, -2$x=5,−2
Q8. Factor = $x^2 – 5x + 6$x2−5x+6
Q9. Quotient = $2x^2 – x – 3$2×2−x−3, Remainder = 12
Q10. Sum = 8, Product = 15
Q11. Root = 2
Q12. $(x + 3)(x^2 – x – 4)$(x+3)(x2−x−4)
Q13. Not a factor
Q14. Quotient = $x^3 – x^2 + 0x + 1$x3−x2+0x+1, Remainder = 0
Q15. $x = 2, -3$x=2,−3
Q16. Remainder = 3
Q17. $(x – 3)(x + 3)$(x−3)(x+3)
Q18. $x = 2$x=2 (repeated root)
Q19. Quotient = $x^2 – 0x – 1$x2−0x−1, Remainder = 0
Q20. Yes
Q21. $(x – 3)(x^2 + 0x – 4)$(x−3)(x2+0x−4)
Q22. Sum = 5/3
Q23. $(x + 3)(x + 4)$(x+3)(x+4)
Q24. Quotient = $x^2 + 5x + 6$x2+5x+6, Remainder = 0
Q25. Factor = $x^2 – 4$x2−4