Learn subjects / Grade 10 / Mathematics GRADE 10 · Mathematics · LEARNING COMPANION
Polynomials Three guided lessons with explanations, visual steps, practice and a final check.
Original learning companion, not full textbook exercise solutions. Three lessons cover zeroes, simple quadratic factorisation and coefficient relations. A full range of graph shapes and every textbook exercise remain outside this companion.
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Keep my work Clear this chapter and space YOUR LEARNING JOURNEY
01 · Get ready 02 · Learn 03 · Practise 04 · Final check 05 · My learning You can revisit any stage. Opening a page does not count as passing. Read all lesson explanations Read all lessons These explanations remain available without JavaScript. The separate learner and solution pages can be printed on A4.
LESSON 1
Zeroes and the x-axis Connect a zero, an input value and an x-intercept.
A polynomial gives an output for each input. A zero is an input whose output is zero.
Evaluate the input For p(x) = (x − 1)(x − 3), substitute a candidate value for x.
Test x = 1 p(1) = (1 − 1)(1 − 3) = 0 × (−2) = 0.
Test x = 3 p(3) = (3 − 1)(3 − 3) = 2 × 0 = 0. The graph meets the x-axis at 1 and 3.
Separate input and point The zeroes are the numbers 1 and 3. The corresponding points are (1, 0) and (3, 0). The sketch is schematic.
Worked example: For p(x) = x² − 4x + 3, p(1) = 1 − 4 + 3 = 0 and p(3) = 9 − 12 + 3 = 0. Thus 1 and 3 are zeroes; p(0) = 3 is the y-intercept.
Watch for this: A zero is an x-value, not the y-intercept. A graph can touch the x-axis without crossing it.
Discuss: Choose (x − 2)(x − 5). Make a table for x = 0 through 6 and mark the two zero outputs.
LESSON 2
Factor a quadratic to find its zeroes Use multiplication to connect factors with zeroes.
Expanding (x − a)(x − b) gives x² − (a + b)x + ab.
Choose a product Consider x² − 7x + 12. We need two numbers with sum 7 and product 12.
Find the pair The pair 3 and 4 works because 3 + 4 = 7 and 3 × 4 = 12.
Check by expansion (x − 3)(x − 4) = x² − 4x − 3x + 12 = x² − 7x + 12.
Set each factor to zero x − 3 = 0 or x − 4 = 0, so the zeroes are 3 and 4.
Worked example: For x² + x − 6, the numbers 3 and −2 have sum 1 and product −6. Thus p(x) = (x + 3)(x − 2), so its zeroes are −3 and 2.
Watch for this: Factoring is not dividing the whole expression by x. Always expand your proposed factors to check the signs.
Discuss: Create a polynomial with zeroes −2 and 5. Expand (x + 2)(x − 5) and verify both inputs.
LESSON 3
Relate zeroes to coefficients Check the sum and product of two zeroes without solving again.
A nonzero leading coefficient changes coefficient ratios; it does not allow you to ignore a.
Start from factors If α and β are zeroes, write p(x) = a(x − α)(x − β).
Expand the structure Expansion gives ax² − a(α + β)x + aαβ.
Compare coefficients Matching ax² + bx + c gives b = −a(α + β) and c = aαβ.
Divide by a Since a ≠ 0, α + β = −b/a and αβ = c/a.
Worked example: For 2x² − 10x + 12, a = 2, b = −10 and c = 12. Sum = 10/2 = 5 and product = 12/2 = 6. Zeroes 2 and 3 satisfy both checks.
Watch for this: The sum has a minus sign; the product does not. Both relations divide by a.
Discuss: Write a monic quadratic whose zeroes have sum 9 and product 14. Explain why x² − 9x + 14 works.
Continue on paper Sources and coverage Original learning companion, not full textbook exercise solutions. Three lessons cover zeroes, simple quadratic factorisation and coefficient relations. A full range of graph shapes and every textbook exercise remain outside this companion.
Original HelioLearn teaching and questions. Source-aligned companion; not an official NCERT/CBSE publication or full textbook solution bank. Independent teacher review and narrated video are pending.
