Polynomials
CBSE Class 10 · Mathematics · Notes, formulas and practice questions
Learn what a zero of a polynomial means, how to find zeroes from the x-intercepts of its graph, and how the zeroes of a quadratic polynomial are tied to its coefficients — including how to form a quadratic from its given zeroes.
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In this chapter, a zero of a polynomial p(x) is a real number a for which p(a) = 0. Geometrically, those zeroes are exactly the x-coordinates of the points where the graph y = p(x) touches or crosses the x-axis. For a linear polynomial the graph is a straight line, so there is always exactly one zero. For a quadratic polynomial the graph is a parabola; it may cut the x-axis at two points, touch it at just one point, or never meet it, giving two, one (repeated), or no real zeroes respectively.
The central algebraic fact for quadratics is the connection between the zeroes and the three coefficients. If ax² + bx + c has zeroes α and β, then α + β = −b/a and αβ = c/a. These two formulas let you calculate the sum and product without first factorising or solving the equation. They are also the key to verifying whether a pair of values really are the zeroes, and to solving for an unknown coefficient when partial information about the zeroes is known.
It is just as useful to work in the opposite direction. When the two zeroes α and β are known, a quadratic polynomial that has them is k[x² − (α + β)x + αβ], where k is any non-zero constant. Choosing k = 1 gives the simplest monic polynomial x² − (α + β)x + αβ. Scaling k merely multiplies every coefficient by the same number and does not change the zeroes. This is the standard method for writing a quadratic from its zeroes, whether the zeroes are integers or fractions.
Graphically, counting the number of points where the curve intersects the x-axis tells you how many real zeroes the polynomial has. A common trap is treating a tangent touch (one point of intersection) as if the zero did not exist; in fact it corresponds to two equal real zeroes. Another is mixing up signs: α + β = −b/a, while αβ = c/a. Always check that the coefficient of x² is not zero and that you are applying these relations only to a quadratic polynomial.
Key terms
- Zero of a polynomial
- A real number a is a zero of the polynomial p(x) if p(a) = 0. A zero is also called a root, and it is the x-coordinate of a point where the graph of y = p(x) meets the x-axis.
- Linear polynomial
- A polynomial of the form ax + b, with a ≠ 0. Its graph is a straight line, and it has exactly one zero, x = −b/a.
- Quadratic polynomial
- An expression of the form ax² + bx + c, with a ≠ 0. Its graph is a parabola: it opens upward when a > 0 and downward when a < 0.
- Parabola and the x-axis
- The zeroes of a quadratic are the x-coordinates of the points where its parabola intersects the x-axis. Two intersections mean two real zeroes, one point of contact means two equal real zeroes, and no intersection means no real zeroes.
- Sum of zeroes
- For a quadratic polynomial ax² + bx + c with zeroes α and β, the sum is α + β = −b/a. This holds regardless of whether the zeroes are rational, irrational, or equal.
- Product of zeroes
- For the same quadratic polynomial, the product of the zeroes is αβ = c/a. Together with the sum, this relation lets you recover coefficients or construct polynomials.
- General quadratic polynomial from its zeroes
- If α and β are the required zeroes, then for any non-zero constant k, the polynomial k[x² − (α + β)x + αβ] has exactly those zeroes. With k = 1, it becomes the monic polynomial x² − (α + β)x + αβ.
Formula sheet
| What | Formula | Notes |
|---|---|---|
| Sum of zeroes of a quadratic | α + β = −b / a | For p(x) = ax² + bx + c with a ≠ 0, α and β are its two zeroes. |
| Product of zeroes of a quadratic | αβ = c / a | α and β are the zeroes of p(x) = ax² + bx + c, with a ≠ 0. |
| Quadratic polynomial from known zeroes | p(x) = k[x² − (α + β)x + αβ] | α and β are the given zeroes; k is any non-zero constant, often taken as 1 to give the monic polynomial. |
Practice questions with answers
1. Find the zeroes of the quadratic polynomial x² − 5x + 6 and verify the relationship between the zeroes and the coefficients.
Factorise: x² − 5x + 6 = (x − 2)(x − 3), so the zeroes are 2 and 3. Their sum is 2 + 3 = 5, which equals −b/a = −(−5)/1 = 5; their product is 2 × 3 = 6, which equals c/a = 6/1 = 6. Thus the relationships are verified.
2. If α and β are zeroes of 3x² − 2x − 5, find α + β and αβ.
Here a = 3, b = −2, c = −5. Therefore α + β = −b/a = −(−2)/3 = 2/3, and αβ = c/a = −5/3.
3. Find a quadratic polynomial whose zeroes are −4 and 3/2.
Sum of zeroes = −4 + 3/2 = −8/2 + 3/2 = −5/2; product = (−4) × (3/2) = −6. A polynomial is k[x² − (−5/2)x + (−6)] = k[x² + (5/2)x − 6]. Taking k = 1 gives x² + (5/2)x − 6, or equivalently 2x² + 5x − 12 after clearing the fraction.
4. The graph of the quadratic polynomial p(x) = x² + bx + c has x-intercepts at (3,0) and (−1,0). Find b and c.
The zeroes are therefore 3 and −1. Their sum is 3 + (−1) = 2 = −b/1, giving b = −2. Their product is 3 × (−1) = −3 = c/1, giving c = −3. Hence p(x) = x² − 2x − 3.
5. One zero of the polynomial p(x) = x² − 2x + k is 3. Find the other zero and the value of k.
The sum of zeroes is −b/a = −(−2)/1 = 2. If the other zero is r, then 3 + r = 2, so r = −1. Also the product equals k, so k = 3 × (−1) = −3. Therefore the other zero is −1 and k = −3.
6. The graph of a linear polynomial is a straight line that crosses the x-axis at (−6,0). What is its only zero?
The zero of a polynomial is the x-coordinate of the point where its graph meets the x-axis. Therefore the only zero is −6.
7. Write a quadratic polynomial whose zeroes have sum 5 and product 6.
For a monic quadratic, p(x) = x² − (sum)x + product = x² − 5x + 6. More generally, p(x) = k(x² − 5x + 6) for any non-zero constant k, which scales the coefficients but leaves the zeroes unchanged.
8. Can a quadratic polynomial have three distinct real zeroes? Explain.
No. The graph of a quadratic is a parabola, and a parabola can intersect the x-axis in at most two points. Algebraically, a degree‑2 equation p(x) = 0 has at most two distinct solutions, so three distinct zeroes are impossible.
9. A parabola opens upward and its lowest point is above the x-axis. How many real zeroes does the corresponding quadratic polynomial have?
When the vertex is above the x-axis and the arms open upward, the whole curve lies above the x-axis. It never touches or crosses the axis, so the polynomial has no real zeroes.
10. If α and β are zeroes of 2x² − 6x + 4, form a quadratic polynomial whose zeroes are α + β and αβ.
From the coefficients, α + β = −(−6)/2 = 3 and αβ = 4/2 = 2. So the new zeroes are 3 and 2. Their sum is 3 + 2 = 5 and their product is 3 × 2 = 6, giving the polynomial x² − 5x + 6 (or any non-zero constant multiple).
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