Polynomials
Board: CBSE |Class:Class10
ComprehensivestudynotesforPolynomialsbyAjayYadav(MathKingofKatargam).Mastereveryconceptwithclearexplanations,solvedexamples,andpracticeproblems.
Thedegreeofapolynomialisthehighestexponentofthevariable.Degree1:Linear(e.g.,ax+b).Degree2:Quadratic(e.g.,ax²+bx+c).Degree3:Cubic(e.g.,ax³+bx²+cx+d).
Azero(root)ofp(x)isavalueasuchthatp(a)=0.Graphically,zerosarethex-coordinateswherethegraphcrossesthex-axis.Alinearpolynomialhas1zero.Aquadratichasatmost2zeros.Acubichasatmost3zeros.
Forquadraticp(x)=ax²+bx+c:Sumofzeros=-b/a.Productofzeros=c/a.Forcubic:Sum=-b/a.Sumofpairwiseproducts=c/a.Product=-d/a.
Ifp(x)andg(x)arepolynomialswithdeg(p)≥deg(g),thenp(x)=g(x)×q(x)+r(x),wheredeg(r)<deg(g).
Thegraphofalinearpolynomialisastraightline.Aquadraticpolynomialgivesaparabola(opensupwardifa>0,downwardifa<0).Acubicpolynomialhasacurvedshapewithatmosttwoturningpoints.
| Quadraticzeros | Sum=-b/a,Product=c/a |
| Cubiczeros | Sum=-b/a,Sum(pairwise)=c/a,Product=-d/a |
| QuadraticFormula | x=(-b±√(b²-4ac))/2a |
| DivisionAlgorithm | p(x)=g(x)q(x)+r(x) |
| Discriminant | D=b²-4ac(D>0:2realroots,D=0:1root,D<0:norealroots) |
Example1:Find thezerosofx²-3x+2.
Solution:x²-3x+2=(x-1)(x-2).Zeros:x=1andx=2.
Example2:Find aquadraticwhosesumofzeros=5andproduct=6.
Solution:x²-(sum)x+product=x²-5x+6.Answer:x²-5x+6.
Example3:Dividex³-3x²+3x-1byx-1.
Solution:Usingsyntheticdivision:1| 1 -3 3 -1. Bring 1. 1+(-3)=-2. -2+3=1. 1+(-1)=0. Quotient: x² – 2x + 1. Remainder: 0.
Practice Questions
- Find the zeros of 2x² – 7x + 3.
- Verify the relationship between zeros and coefficients for x² – 5x + 6.
- Find a cubic with zeros 1, 2, 3.
- Divide 3x³ + 4x – 1 by x + 1.
- If one zero of 2x² – 5x + k is 3, find k and the other zero.
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