Polynomials – important questions for Boards 2027
CBSE Class 10 Maths
How many zeroes at most can a polynomial of degree n have?
Answer
A polynomial p(x) of degree n has at most n zeroes, since its graph y = p(x) intersects the x-axis at at most n points.
Define the degree of a non-zero polynomial in one variable.
Answer
The degree of a non-zero polynomial in one variable is the highest power of its variable with a non-zero coefficient. For example, 5x³ − 4x² + x − 2 has degree 3.
If α, β and γ are the zeroes of ax³ + bx² + cx + d, where a ≠ 0, state the relations for (i) α + β + γ, (ii) αβ + βγ + γα, and (iii) αβγ.
Answer
1. For ax³ + bx² + cx + d, where a ≠ 0, with zeroes α, β and γ, their sum is α + β + γ = −b/a. 2. The sum of products taken two at a time is αβ + βγ + γα = c/a. 3. The product of the three zeroes is αβγ = −d/a, the negative ratio of constant term d to leading coefficient a.
(5 marks) (a) State the general form of a cubic polynomial in x and the conditions on its coefficients. (b) Explain how p(k) is defined for a polynomial p(x). (c) State the condition that a real number k must satisfy to be a zero of p(x).
Answer
Polynomials are classified by their degree, and their values depend on the number substituted for the variable. 1. The general form of a cubic polynomial in x is ax³ + bx² + cx + d, where a, b, c and d are real numbers and a ≠ 0. The non-zero coefficient a makes the highest power of x equal to 3. 2. The value p(k) is obtained by replacing x by any real number k in p(x). This value is denoted by p(k). 3. A real number k is a zero of p(x) precisely when p(k) = 0. Thus, substitution of k into the polynomial must give zero. The cubic form and the condition p(k) = 0 specify the polynomial’s degree and its zeroes, respectively.
For y = ax² + bx + c, state whether the parabola opens upwards or downwards when a > 0 and when a < 0.
Answer
When a > 0, the parabola opens upwards; when a < 0, it opens downwards.
Name the types of polynomials of degrees 1, 2 and 3, respectively.
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Explain the relationship between the zeroes of a polynomial p(x) and the graph of y = p(x). In your answer, state what the x-coordinates of the graph’s intersections with the x-axis represent.
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For a quadratic polynomial ax² + bx + c with zeroes α and β, write the formula for α + β.
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Write the general form of a quadratic polynomial in x and state the condition on its leading coefficient.
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If the sum and product of the zeroes of a quadratic polynomial are S and P, respectively, which is one such polynomial? (A) x² + Sx + P (B) x² − Sx + P (C) x² + Px − S (D) x² − Px − S
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8 more probable questions in this chapter, every one with a full answer.
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