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CBSE Class 10 · Mathematics

Polynomials

Official NCERT chapter from Mathematics (book code jemh1). ExamMaster notes are original teaching at CBSE Class 10 depth.

This lesson follows the official chapter “Polynomials” in Mathematics. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 10
  • Medium level
  • 8 concepts

1Geometrical Meaning of the Zeroes of a Polynomial

A zero of p(x) is an x that makes p(x) = 0. Geometrically, for a graph y = p(x), zeroes are x-intercepts. x² − 5x + 6 meets the axis at 2 and 3. A graph that misses the axis has no real zero in this picture — that is a look, not a mood.

A pretty dip that never crosses is not a zero.

Figure. Zeroes are where the graph meets the x-axis. For p(x)=(x-1)(x-3) those meetings are x=1 and x=3, not the turning point.

How it works

  1. Set p(x) = 0, or read the x-interceptsThe meaning.
  2. Name those x-valuesThe zeroes.
  3. Refuse a y-intercept as a zero unless it is also on the x-axisGeometry of the heading.

2Relationship between Zeroes and Coefficients of a Polynomial

For a quadratic ax² + bx + c, the sum of zeroes is −b/a and the product is c/a (as taught). For x² − 5x + 6, sum 5, product 6 — matching 2 and 3. The relation is a check and a build; it is not a third zero.

Using +b/a for the sum is the sign-steal.

Figure. For x² − 5x + 4 the zeroes 1 and 4 add to 5 = −b/a and multiply to 4 = c/a. Sum and product read the coefficients; they are not a new pair of numbers.

How it works

  1. Read a, b, cThe coefficients.
  2. Write sum = −b/a, product = c/aThe relation.
  3. Check against the intercepts if you have themHonest pair.

x² − 5x + 6

Find the zeroes of x² − 5x + 6 and check sum and product.

  • Factor(x−2)(x−3)
  • Zeroes2 and 3
  • Sum / product5 and 6 = −(−5)/1 and 6/1

Pro tip. Zeroes are intercepts; sum and product reuse a, b, c.

3A definition is a test you can run

Polynomial / zero is a test: an x that makes the writing 0, or an intercept. If you only say “root-ish”, you have a heading.

A doodle through (0, 6) only is not two zeroes.

Figure. A zero is a test you run: feed the input to p. For p(x)=x²−4, p(2)=0 so 2 is a zero; p(3)=5 so 3 is not. The name waits for the output 0.

How it works

  1. Name p(x) and the x you claimThe object.
  2. Evaluate or read the interceptThe test.
  3. Then the word has contentThe definition ran.

4Name the given before the unknown

The given is the writing or the graph. The unknown is a zero or the sum/product. Copy a, b, c before you flip a sign.

Reading 5x as +5 in −b/a without the minus on b is a silent given.

Figure. Name a, b, c of ax²+bx+c before you hunt the zeroes. For 1, −5, 4 the unknown pair is not free to invent — it must fit those three given numbers.

How it works

  1. Copy the polynomialThe given.
  2. Name zeroes or the pair of relationsThe unknown.
  3. Then factor or use −b/aGiven first.

5One worked case is enough at this class

One quadratic is enough. A page of the same x²−5x+6 clone does not add a new intercept-meaning.

Ten twins of 2 and 3 are still one idea.

Figure. One worked quadratic is enough at this class: x²−5x+6 has zeroes 2 and 3. A second example is not a new rule.

How it works

  1. Find one pair of zeroes or one sum/productThe case.
  2. A second only as a checkOptional.
  3. StopThis class.

6A miss: swapping the school name for the picture

A roller-coaster photo is a setting. The maths is y = p(x) and the x-intercepts. Hills are not zeroes.

A skyline is not a polynomial.

Figure. The turning point is a picture of the lowest (or highest) place. A zero is an axis cut. Calling the vertex a zero swaps the school name for the picture.

How it works

  1. Keep the photo as a storyA setting.
  2. Write p(x) and read interceptsThe maths.
  3. Then sum and productDo not swap.

7Check by the opposite action or the opposite test

Check zeroes by substituting back: p(2) and p(3) must be 0. Check the relation by adding and multiplying the zeroes against −b/a and c/a.

A graph that “looks like it crosses at 2” without a substitute is unchecked.

Figure. Check zeroes by the opposite move: expand (x−2)(x−3) and demand x²−5x+6, the polynomial you started with. A new pair that does not rebuild p has failed the test.

How it works

  1. State 2 and 3, or sum 5The claim.
  2. Substitute or add/multiplyThe check.
  3. Mend if they disagreeHonest.

8Keep the claim at this chapter, not the next

This chapter’s claim stops at zeroes and the quadratic relations. It does not steal a pair of linear graphs, and it does not start the quadratic-formula chapter’s discriminant lecture beyond what a zero already is.

A page that already “eliminates” two equations here misses the pair-chapter.

Figure. This chapter claims real axis meetings you can plot. Complex zeroes are a later chapter’s claim — do not spend this page on them.

How it works

  1. Keep intercepts and −b/a, c/aThis chapter.
  2. Leave two-line graphs for the pair chapterThe next.
  3. Do not treat the map as already answeredClaim-size.
For x² − 5x + 6 the sum of zeroes is
  1. 5
  2. −5
  3. 6

−b/a = 5; product is 6.

Notes

  • The official chapter title is “Polynomials”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • zero ⇔ p(x)=0 ⇔ x-intercept of y=p(x)
  • quadratic: sum = −b/a, product = c/a

Recap

Hold these pegs from the official chapter “Polynomials”. The wording is ExamMaster’s teaching, not a textbook recap.

Geometrical Meaning of the Zeroes of a Polynomial
A zero of p(x) is an x that makes p(x) = 0.
Relationship between Zeroes and Coefficients of a Polynomial
For a quadratic ax² + bx + c, the sum of zeroes is −b/a and the product is c/a (as taught).
A definition is a test you can run
Polynomial / zero is a test: an x that makes the writing 0, or an intercept.
Name the given before the unknown
The given is the writing or the graph. The unknown is a zero or the sum/product. Copy a, b, c before you flip a sign.
One worked case is enough at this class
One quadratic is enough. A page of the same x²−5x+6 clone does not add a new intercept-meaning.
A miss: swapping the school name for the picture
A roller-coaster photo is a setting. The maths is y = p(x) and the x-intercepts. Hills are not zeroes.

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