Polynomials — Class 10 MCQs with Answers

Polynomials — important questions & MCQs with answers (Class 10 Mathematics)

6 solved questions from this chapter's difficulty levels, each with its answer and explanation. Tap an option to check yourself.

0 of 6 answered

1. Q1 of 6Easy

   The degree of the polynomial p(x) = 4x³ − 2x² + 7x − 5 is:

   A1B2C3D5
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   Answer: 3

   Explanation: Degree = highest power of x with a nonzero coefficient. In 4x³ − 2x² + 7x − 5 that's x³, so the degree is 3, not the number of terms (4).

2. Q2 of 6Easy

   The degree of the polynomial 4x³ − 2x² + 5x − 7 is:

   A3B2C1D4
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   Answer: 3

   Explanation: Degree is the highest exponent of x with a nonzero coefficient. In 4x³ − 2x² + 5x − 7 the leading term is 4x³, giving degree 3.

3. Q3 of 6Medium

   If one zero of x² − 5x + k is 2, then k equals:

   A6B−6C10D−10
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   Answer: 6

   Explanation: Substitute the known zero into the polynomial: p(2) = 2² − 5(2) + k = 4 − 10 + k = 0, so k = 6.

4. Q4 of 6Medium

   Assertion (A): The quadratic polynomial x² – 5x + 6 has no zeroes. Reason (R): The zeroes of a polynomial p(x) are the x-coordinates of the points where the graph of y = p(x) intersects the x-axis.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: A is false but R is true

   Explanation: R is true. A is false: x² – 5x + 6 = (x – 2)(x – 3), so 2 and 3 are its zeroes, and its graph cuts the x-axis at x = 2 and x = 3.

5. Q5 of 6Hard

   If α and β are zeroes of x² − 5x + 6, then α² + β² =

   A37B25C13D12
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   Answer: 13

   Explanation: α + β = −(−5)/1 = 5 and αβ = 6/1 = 6. Using α² + β² = (α+β)² − 2αβ = 25 − 12 = 13; forgetting to subtract 2αβ gives the tempting 25 instead.

6. Q6 of 6Hard

   Assertion (A): The zeroes of 4x² – 4x – 3 are 3/2 and –1/2. Reason (R): The graph of y = ax² + bx + c opens upwards when a > 0.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: Both A and R are true but R is not the correct explanation of A

   Explanation: A is true: 4x² – 4x – 3 = (2x – 3)(2x + 1), which is 0 at x = 3/2 and x = –1/2. R is also true, but the shape of the graph does not tell us these zeroes.

Polynomials notes: key concepts (Class 10 Mathematics)

A polynomial's zeroes are where its graph meets the x-axis, and they are tied to its coefficients by fixed relationships. This chapter reads zeroes off a graph, links them to the coefficients for quadratics and cubics, and divides one polynomial by another.

Polynomial and its degree

An expression in whole-number powers of a variable; the degree is the highest power present, and it fixes the maximum number of zeroes.

Linear, quadratic and cubic

Degree 1, 2 and 3 polynomials, having at most 1, 2 and 3 zeroes respectively — the degree caps the number of zeroes.

Zero of a polynomial

A value of x for which p(x) = 0. Geometrically it is the x-coordinate of a point where the graph crosses the x-axis.

Zeroes from a graph

The number of times the curve cuts the x-axis equals the number of real zeroes; a curve that misses the axis has none.

Sum and product of zeroes (quadratic)

For ax² + bx + c the zeroes add to −b/a and multiply to c/a, so a quadratic can be built from its zeroes.

Forming a quadratic from its zeroes

Given the sum s and product p of the zeroes, the polynomial is k[x² − sx + p] for any non-zero k.

Zeroes of a cubic

For ax³ + bx² + cx + d the zeroes satisfy α+β+γ = −b/a, αβ+βγ+γα = c/a and αβγ = −d/a.

Division algorithm for polynomials

For polynomials p(x) and g(x) ≠ 0, p(x) = g(x)·q(x) + r(x), where the remainder r(x) has degree less than g(x).

Long division of polynomials

Divide the leading terms, multiply back, subtract and repeat — arranging both polynomials in descending powers first.

Using a known zero to factorise

If α is a zero then (x − α) is a factor, so dividing by it reduces a cubic to a quadratic that can be solved normally.