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EAPCET Mathematics · Intermediate second year · Paper IIA

Theory of Equations

Theory of Equations extends quadratics to higher-degree polynomials: relations between roots and coefficients, transformed equations and reciprocal equations.

Subject
Mathematics
Taught in
Intermediate second year · Paper IIA
Mathematics in the paper
80 of 160 questions

Key formulas and results

  • For ax3+bx2+cx+d=0ax^{3} + bx^{2} + cx + d = 0: ∑α=−ba\sum\alpha = -\dfrac{b}{a}, ∑αβ=ca\sum\alpha\beta = \dfrac{c}{a}, αβγ=−da\alpha\beta\gamma = -\dfrac{d}{a}
  • For polynomials with real coefficients, complex roots occur in conjugate pairs
  • Roots increased by hh: replace xx with x−hx - h
  • Roots multiplied by kk: replace xx with xk\dfrac{x}{k}
  • Reciprocal roots: replace xx with 1x\dfrac{1}{x}
  • Cubic with roots α,β,γ\alpha, \beta, \gamma: x3−S1x2+S2x−S3=0x^{3} - S_{1}x^{2} + S_{2}x - S_{3} = 0, where S1=∑αS_{1} = \sum\alpha, S2=∑αβS_{2} = \sum\alpha\beta, S3=αβγS_{3} = \alpha\beta\gamma
  • For ax4+bx3+cx2+dx+e=0ax^{4} + bx^{3} + cx^{2} + dx + e = 0: ∑α=−ba\sum\alpha = -\tfrac{b}{a}, ∑αβ=ca\sum\alpha\beta = \tfrac{c}{a}, ∑αβγ=−da\sum\alpha\beta\gamma = -\tfrac{d}{a}, αβγδ=ea\alpha\beta\gamma\delta = \tfrac{e}{a}
  • For anxn+⋯+a1x+a0=0a_{n}x^{n} + \dots + a_{1}x + a_{0} = 0: sum of roots =−an−1an= -\dfrac{a_{n-1}}{a_{n}}, product of roots =(−1)na0an= (-1)^{n}\dfrac{a_{0}}{a_{n}}
  • Roots squared: replace xx with x\sqrt{x}; roots cubed: replace xx with x3\sqrt[3]{x}; roots aα+ba\alpha + b: replace xx with x−ba\dfrac{x - b}{a}

What this chapter covers

  • Relations between roots and coefficients
  • Roots in arithmetic, geometric or harmonic progression
  • Transformation of equations
  • Reciprocal equations
  • Multiple roots

How it is usually asked

  • Find Σα², Σ1/α and similar sums from coefficients
  • Solve an equation whose roots are in A.P. or G.P.
  • Find a transformed equation

Understanding Theory of Equations

A polynomial equation of degree n has exactly n roots, counted with multiplicity, and they can be real or complex. The relations between roots and coefficients extend those of quadratics: for ax³ + bx² + cx + d = 0, Σα = −b/a, Σαβ = c/a and αβγ = −d/a. The signs alternate, starting with a minus for the sum, and the same pattern continues for higher degrees.

These relations let you compute symmetric functions of the roots without solving. For example Σα² = (Σα)² − 2Σαβ and Σ1/α = Σαβ/αβγ. They also solve equations when the roots have a special pattern: taking roots in A.P. as a − d, a, a + d makes the sum give a at once, and taking roots in G.P. as a/r, a, ar makes the product give a.

A transformed equation has roots related to the old ones by a simple rule. To get roots increased by h, replace x with x − h; multiplied by k, replace x with x/k; reciprocals, replace x with 1/x. If a polynomial has real coefficients, complex roots come in conjugate pairs, and with rational coefficients, roots like p + √q come in pairs too. EAPCET tests all of these directly.

Worked example

EAPCET-style question

The roots of x3−9x2+23x−15=0x^{3} - 9x^{2} + 23x - 15 = 0 are in arithmetic progression. The roots are

  1. (1)1, 3, 5
  2. (2)2, 3, 4
  3. (3)−1, 3, 7
  4. (4)1, 4, 7

Solution

  1. Take the roots as a−d,a,a+da - d, a, a + d. Their sum 3a=93a = 9, so a=3a = 3.
  2. Product: (3−d)(3)(3+d)=15(3 - d)(3)(3 + d) = 15, so 9−d2=59 - d^{2} = 5 and d=±2d = \pm 2.
  3. The roots are 1, 3, 5.
  4. Check the middle relation: 1⋅3+3⋅5+5⋅1=231 \cdot 3 + 3 \cdot 5 + 5 \cdot 1 = 23, which matches.

Answer: 1, 3, 5

More practice questions

Practice question 1

The cubic equation whose roots are 1, 2 and 3 is

  1. (1)x3−6x2+11x−6=0x^{3} - 6x^{2} + 11x - 6 = 0
  2. (2)x3+6x2+11x+6=0x^{3} + 6x^{2} + 11x + 6 = 0
  3. (3)x3−6x2−11x−6=0x^{3} - 6x^{2} - 11x - 6 = 0
  4. (4)x3−11x2+6x−6=0x^{3} - 11x^{2} + 6x - 6 = 0

Solution

  1. S1=1+2+3=6S_{1} = 1 + 2 + 3 = 6
  2. S2=1⋅2+2⋅3+3⋅1=11S_{2} = 1 \cdot 2 + 2 \cdot 3 + 3 \cdot 1 = 11
  3. S3=1⋅2⋅3=6S_{3} = 1 \cdot 2 \cdot 3 = 6
  4. x3−S1x2+S2x−S3=x3−6x2+11x−6=0x^{3} - S_{1}x^{2} + S_{2}x - S_{3} = x^{3} - 6x^{2} + 11x - 6 = 0

Answer: x3−6x2+11x−6=0x^{3} - 6x^{2} + 11x - 6 = 0

Practice question 2

If α\alpha and β\beta are the roots of x2−5x+6=0x^{2} - 5x + 6 = 0, the equation whose roots are α2\alpha^{2} and β2\beta^{2} is

  1. (1)x2−13x+36=0x^{2} - 13x + 36 = 0
  2. (2)x2−25x+36=0x^{2} - 25x + 36 = 0
  3. (3)x2+13x+36=0x^{2} + 13x + 36 = 0
  4. (4)x2−13x+6=0x^{2} - 13x + 6 = 0

Solution

  1. Replace xx with x\sqrt{x}: x−5x+6=0x - 5\sqrt{x} + 6 = 0
  2. x+6=5xx + 6 = 5\sqrt{x}; squaring gives x2+12x+36=25xx^{2} + 12x + 36 = 25x
  3. x2−13x+36=0x^{2} - 13x + 36 = 0 (check: roots 4 and 9 are 222^{2} and 323^{2})

Answer: x2−13x+36=0x^{2} - 13x + 36 = 0

Common mistakes to avoid

  • Writing αβγ = d/a for a cubic; the product of roots of ax³ + bx² + cx + d = 0 is −d/a.
  • Replacing x with x + h to increase roots by h; that decreases them, and the correct substitution is x − h.
  • Pairing irrational roots p + √q with p − √q when the coefficients are only real; the pairing is guaranteed only for rational coefficients.
  • Computing Σα² as (Σα)² without subtracting 2Σαβ.

How to prepare it

For roots in A.P. take a − d, a, a + d; for G.P. take a/r, a, ar — the sum or product then gives a directly.

Once the formulas feel familiar, test them the way the exam does: practise Theory of Equations questions chapter-wise, then attempt a full previous year paper to see whether it holds up under a 3-hour timer.

Related Mathematics chapters

See every chapter in the EAPCET Mathematics syllabus.

Practise Theory of Equations chapter-wise

Chapter-wise quizzes show exactly which topics in this chapter you still get wrong — before a full paper does.

Frequently asked questions

Is Theory of Equations part of the EAPCET Mathematics syllabus?
Yes. Theory of Equations is taught in Intermediate second year · Paper IIA Mathematics, and the EAPCET Engineering syllabus is based on the Intermediate first- and second-year syllabus. Check the official notification each year for any changes.
How should I prepare Theory of Equations for EAPCET?
For roots in A.P. take a − d, a, a + d; for G.P. take a/r, a, ar — the sum or product then gives a directly.
How many Mathematics questions are there in EAPCET?
80 of the 160 questions in the Engineering paper are Mathematics, spread across 37 chapters.
What mistakes do students make in Theory of Equations?
Writing αβγ = d/a for a cubic; the product of roots of ax³ + bx² + cx + d = 0 is −d/a. Replacing x with x + h to increase roots by h; that decreases them, and the correct substitution is x − h. Pairing irrational roots p + √q with p − √q when the coefficients are only real; the pairing is guaranteed only for rational coefficients. Computing Σα² as (Σα)² without subtracting 2Σαβ.