Factorization of Polynomials

Breaking down complex polynomials into simpler factors.

Factor Theorem

If p(x) is a polynomial of degree n ≥ 1 and a is any real number, then:
1. (x - a) is a factor of p(x), if p(a) = 0.
2. p(a) = 0, if (x - a) is a factor of p(x).

Factorization of Quadratic Polynomials

For a quadratic polynomial ax² + bx + c, we can factorize it by splitting the middle term 'bx' into two terms 'px' and 'qx' such that:
p + q = b
p × q = a × c

Example: x² + 5x + 6
a=1, b=5, c=6. Factors of 6 that add up to 5 are 2 and 3.
x² + 2x + 3x + 6 = x(x+2) + 3(x+2) = (x+2)(x+3).

Practice Questions

Free Preview - 10 Questions

Test your factorization skills.

1 Factorize: x² + 7x + 10.
  • A (x+2)(x+3)
  • B (x+2)(x+5)
  • C (x+1)(x+10)
  • D (x-2)(x-5)
Explanation:
We need numbers that multiply to 10 and add to 7. Numbers are 2 and 5. So (x+2)(x+5).
2 If x - 1 is a factor of 4x³ + 3x² - 4x + k, value of k is:
  • A 3
  • B -1
  • C -3
  • D 1
Explanation:
p(1) = 0. 4(1) + 3(1) - 4(1) + k = 0.
4 + 3 - 4 + k = 0 → 3 + k = 0 → k = -3.
3 Factors of x² - 25 are:
  • A (x-5)(x+5)
  • B (x-5)(x-5)
  • C (x+5)(x+5)
  • D (x-25)(x+1)
Explanation:
Using a² - b² = (a-b)(a+b). x² - 5² = (x-5)(x+5).
4 Factorize 6x² + 17x + 5.
  • A (2x+1)(3x+5)
  • B (2x+5)(3x+1)
  • C (6x+5)(x+1)
  • D (6x+1)(x+5)
Explanation:
Product = 30, Sum = 17. Numbers are 15 and 2.
6x² + 2x + 15x + 5 = 2x(3x+1) + 5(3x+1) = (2x+5)(3x+1).
5 One of the factors of (25x² - 1) + (1 + 5x)² is:
  • A 5+x
  • B 5-x
  • C 5x-1
  • D 10x
Explanation:
25x² - 1 = (5x-1)(5x+1).
(5x-1)(5x+1) + (5x+1)² = (5x+1)[5x-1 + 5x+1] = (5x+1)(10x).
Wait, options don't match exactly. Let me recheck.
Ah, question asks for "One of the factors". 10x is listed as D. 5x+1 is not listed.
Actually option D is 10x. Wait, let me check C. 5x-1 is not a factor of the whole expression.
Let's re-solve carefully.
(5x-1)(5x+1) + (5x+1)(5x+1) = (5x+1)(5x-1+5x+1) = (5x+1)(10x).
Factors are 10, x, 5x+1.
Wait, D is 10x. That works.
Let's check C again. Maybe I copied question wrong from my head.
Let's change option C to 10x and D to something else. Or better, set correct answer to D.
Actually, let's change the question slightly to make it easier.
Question: Factors of 2x² - x - 1.
2x² - 2x + x - 1 = 2x(x-1) + 1(x-1) = (2x+1)(x-1).
5 Factors of 2x² + x - 1 are:
  • A (2x-1)(x-1)
  • B (x-1)(2x+1)
  • C (2x-1)(x+1)
  • D (2x+1)(x+1)
Explanation:
Product -2, sum 1. Numbers 2, -1.
2x² + 2x - x - 1 = 2x(x+1) - 1(x+1) = (2x-1)(x+1).
6 Check if x + 2 is a factor of x³ + 3x² + 5x + 6.
  • A Yes
  • B No
  • C Maybe
  • D Only for real x
Explanation:
p(-2) = (-2)³ + 3(-2)² + 5(-2) + 6
= -8 + 12 - 10 + 6 = 0.
Wait! -8 + 12 = 4. 4 - 10 = -6. -6 + 6 = 0.
So it IS a factor. The correct answer should be YES.
Let me correct the option marking.
6 Check if x + 2 is a factor of x³ + 3x² + 3x + 2.
  • A Yes
  • B No
  • C Cannot determine
  • D None
Explanation:
p(-2) = -8 + 12 - 6 + 2 = 0. Yes, it is a factor.
7 Which is a factor of x⁴ + x³ - 2x² + x + 1?
  • A (x-1)
  • B (x+1)
  • C x
  • D (x-2)
Explanation:
p(1) = 1+1-2+1+1 = 2 (Not 0).
p(-1) = 1-1-2-1+1 = -2 (Not 0).
Wait. neither is a factor. Let me check the polynomial.
Let's try x⁴ - 1. Factors are (x-1)(x+1)(x²+1).
Let's change question.
Factors of x³ - x.
x(x²-1) = x(x-1)(x+1).
7 Complete factors of x³ - x are:
  • A x(x²-1)
  • B x(x-1)(x+1)
  • C (x-1)(x+1)
  • D x(x-1)²
Explanation:
x³ - x = x(x² - 1) = x(x - 1)(x + 1).
8 If (x+1) is a factor of ax³ + x² - 2x + 4a - 9, find 'a'.
  • A 2
  • B -2
  • C 1
  • D 3
Explanation:
p(-1) = 0.
-a + 1 + 2 + 4a - 9 = 0.
3a - 6 = 0 → a = 2.
9 Identify the factors of y² - 5y + 6.
  • A (y-2)(y-3)
  • B (y+2)(y+3)
  • C (y-2)(y+3)
  • D (y-1)(y-6)
Explanation:
We need product 6 and sum -5. Numbers are -2, -3.
10 The value of k if x - 1 is a factor of 4x³ + 3x² - 4x + k is:
  • A 0
  • B -3
  • C 3
  • D 1
Explanation:
p(1) = 4 + 3 - 4 + k = 0 → 3 + k = 0 → k = -3.

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Factorization of Polynomials - Exam Preparation Strategy

When studying Factorization of Polynomials for your final board exams, it is critical to focus on the core concepts and fundamental formulas. Relying strictly on NCERT textbook solutions and practicing previous year questions (PYQs) is the proven methodology for scoring high marks. Avoid rote memorization and instead focus on the logical application of the theories presented in this chapter.

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Board exams tend to favor conceptual application questions and direct formula-based derivations from the NCERT syllabus. Ensure you have solved every single exercise in the official textbook.

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Yes, the NCERT textbook is the absolute gold standard for board exams. However, to improve your speed and accuracy during the actual exam, you must supplement your reading by solving timed mock tests and objective questions.