Polynomials

CBSE Class 10 Maths · 11 questions · 26 marks

In Class 9 you factorised polynomials; now you look at what their zeroes actually tell you. The headline result is that the zeroes of a quadratic are locked to its coefficients by two neat relations, so you can answer questions about α and β without ever finding them. Graphically, a zero is simply a point where the curve crosses the x-axis, which is why a quadratic can have two, one or no real zeroes.

Quick revision

A real number k is a zero of p(x) if p(k) = 0; on the graph, the zeroes are the x-coordinates of the points where the curve meets the x-axis.
A polynomial of degree n has at most n real zeroes: a linear polynomial has 1, a quadratic at most 2, a cubic at most 3.
For a quadratic ax² + bx + c with zeroes α and β: α + β = -b/a and αβ = c/a.
A quadratic with given zeroes can be written as k[x² - (sum of zeroes)x + (product of zeroes)] for any non-zero constant k.
For a cubic ax³ + bx² + cx + d with zeroes α, β, γ: α + β + γ = -b/a, αβ + βγ + γα = c/a and αβγ = -d/a.
The discriminant D = b² - 4ac decides the graph: D > 0 gives two distinct zeroes, D = 0 gives one repeated zero, D < 0 gives no real zeroes.
Useful rearrangements: α² + β² = (α + β)² - 2αβ, and 1/α + 1/β = (α + β)/(αβ).
Also α³ + β³ = (α + β)³ - 3αβ(α + β), and (α - β)² = (α + β)² - 4αβ.
The shape of y = ax² + bx + c is a parabola opening upwards when a > 0 and downwards when a < 0.

Important questions

1
MCQ1 mark
If α and β are the zeroes of the polynomial 3x² - 5x - 2, then α + β equals:
1. A.5/3
2. B.-5/3
3. C.-2/3
4. D.2/3
Answer
α + β = 5/3. For ax² + bx + c the sum of the zeroes is -b/a. Here a = 3, b = -5, c = -2, so α + β = -(-5)/3 = 5/3. As a check, 3x² - 5x - 2 = (3x + 1)(x - 2), whose zeroes are -1/3 and 2, and -1/3 + 2 = 5/3.
2
MCQ1 mark
A quadratic polynomial whose zeroes are 4 and -7 is:
1. A.x² - 3x - 28
2. B.x² + 3x - 28
3. C.x² + 3x + 28
4. D.x² - 3x + 28
Answer
The polynomial is x² + 3x - 28. Sum of zeroes = 4 + (-7) = -3, and product = 4 × (-7) = -28. A quadratic with these zeroes is x² - (sum)x + (product) = x² - (-3)x + (-28) = x² + 3x - 28. Check by factorising: x² + 3x - 28 = (x + 7)(x - 4), giving zeroes -7 and 4.
3
MCQ1 mark
If one zero of the polynomial x² - 8x + k is 3, then the value of k is:
1. A.9
2. B.15
3. C.-15
4. D.24
Answer
k = 15. Since 3 is a zero, substituting x = 3 must give 0: 3² - 8(3) + k = 0. 9 - 24 + k = 0, so k = 15. The polynomial is then x² - 8x + 15 = (x - 3)(x - 5), so the other zero is 5, and indeed 3 + 5 = 8 = -b/a.
4
MCQ1 mark
If α and β are the zeroes of 4x² + 12x + 5, then the value of 1/α + 1/β is:
1. A.12/5
2. B.-12/5
3. C.-5/12
4. D.3/5
Answer
1/α + 1/β = -12/5. Here α + β = -b/a = -12/4 = -3 and αβ = c/a = 5/4. Now 1/α + 1/β = (α + β)/(αβ) = (-3)/(5/4) = -3 × 4/5 = -12/5. As a check, 4x² + 12x + 5 = (2x + 1)(2x + 5), so the zeroes are -1/2 and -5/2, and 1/(-1/2) + 1/(-5/2) = -2 - 2/5 = -12/5.
5
Assertion & Reason1 mark
Assertion (A): The polynomial x² + 6x + 10 has no real zeroes. Reason (R): A quadratic ax² + bx + c has real zeroes only when its discriminant b² - 4ac is greater than or equal to 0.
1. A.Both A and R are true and R is the correct explanation of A
2. B.Both A and R are true but R is not the correct explanation of A
3. C.A is true but R is false
4. D.A is false but R is true
Answer
Both A and R are true and R is the correct explanation of A. Reason R is the standard discriminant test and is true. For x² + 6x + 10, a = 1, b = 6, c = 10, so D = 6² - 4(1)(10) = 36 - 40 = -4. Since D < 0, the polynomial has no real zeroes, so Assertion A is true, and it follows directly from the rule stated in R.
6
Short answer2 marks
Find the zeroes of the polynomial 6x² - x - 2 and verify the relationship between the zeroes and its coefficients.
7
Short answer3 marks
If α and β are the zeroes of the polynomial x² - 7x + 10, find the value of α² + β² and of α/β + β/α.
8
Short answer2 marks
Form a quadratic polynomial with integer coefficients whose zeroes have sum 5/2 and product -3/2.
9
Long answer5 marks
Find the zeroes of the quadratic polynomial 3x² + 5x - 2 and verify the relationship between the zeroes and the coefficients. Hence form a quadratic polynomial with integer coefficients whose zeroes are twice the zeroes of the given polynomial.
10
Long answer5 marks
(a) If α and β are the zeroes of x² - 6x + 8, find the values of 1/α + 1/β and α² + β². (b) Form a quadratic polynomial whose zeroes are (α + 2) and (β + 2).
11
Case study4 marks
Read the following and answer the questions that follow: During a school sports day, a student throws a ball vertically upwards from ground level. Its height above the ground after t seconds is modelled by the polynomial h(t) = -5t² + 20t, where h is measured in metres and t in seconds. (a) Find the zeroes of h(t) and interpret them in the context of the throw. (b) Verify the relationship between the zeroes and the coefficients of h(t). (c) The ball reaches its greatest height midway between the two zeroes. Find that greatest height. (d) Find the height of the ball 3 seconds after it is thrown.

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Polynomials

CBSE Class 10 Maths · 11 questions · 26 marks

Quick revision

A real number k is a zero of p(x) if p(k) = 0; on the graph, the zeroes are the x-coordinates of the points where the curve meets the x-axis.

A polynomial of degree n has at most n real zeroes: a linear polynomial has 1, a quadratic at most 2, a cubic at most 3.

For a quadratic ax² + bx + c with zeroes α and β: α + β = -b/a and αβ = c/a.

A quadratic with given zeroes can be written as k[x² - (sum of zeroes)x + (product of zeroes)] for any non-zero constant k.

For a cubic ax³ + bx² + cx + d with zeroes α, β, γ: α + β + γ = -b/a, αβ + βγ + γα = c/a and αβγ = -d/a.

The discriminant D = b² - 4ac decides the graph: D > 0 gives two distinct zeroes, D = 0 gives one repeated zero, D < 0 gives no real zeroes.

Useful rearrangements: α² + β² = (α + β)² - 2αβ, and 1/α + 1/β = (α + β)/(αβ).

Also α³ + β³ = (α + β)³ - 3αβ(α + β), and (α - β)² = (α + β)² - 4αβ.

The shape of y = ax² + bx + c is a parabola opening upwards when a > 0 and downwards when a < 0.

Important questions

MCQ1 mark

If α and β are the zeroes of the polynomial 3x² - 5x - 2, then α + β equals:

A.5/3
B.-5/3
C.-2/3
D.2/3

MCQ1 mark

A quadratic polynomial whose zeroes are 4 and -7 is:

A.x² - 3x - 28
B.x² + 3x - 28
C.x² + 3x + 28
D.x² - 3x + 28

MCQ1 mark

If one zero of the polynomial x² - 8x + k is 3, then the value of k is:

A.9
B.15
C.-15
D.24

MCQ1 mark

If α and β are the zeroes of 4x² + 12x + 5, then the value of 1/α + 1/β is:

A.12/5
B.-12/5
C.-5/12
D.3/5

Assertion & Reason1 mark

Assertion (A): The polynomial x² + 6x + 10 has no real zeroes. Reason (R): A quadratic ax² + bx + c has real zeroes only when its discriminant b² - 4ac is greater than or equal to 0.

A.Both A and R are true and R is the correct explanation of A
B.Both A and R are true but R is not the correct explanation of A
C.A is true but R is false
D.A is false but R is true

Short answer2 marks

Find the zeroes of the polynomial 6x² - x - 2 and verify the relationship between the zeroes and its coefficients.

Short answer3 marks

If α and β are the zeroes of the polynomial x² - 7x + 10, find the value of α² + β² and of α/β + β/α.

Short answer2 marks

Form a quadratic polynomial with integer coefficients whose zeroes have sum 5/2 and product -3/2.

Long answer5 marks

Find the zeroes of the quadratic polynomial 3x² + 5x - 2 and verify the relationship between the zeroes and the coefficients. Hence form a quadratic polynomial with integer coefficients whose zeroes are twice the zeroes of the given polynomial.

Long answer5 marks

(a) If α and β are the zeroes of x² - 6x + 8, find the values of 1/α + 1/β and α² + β². (b) Form a quadratic polynomial whose zeroes are (α + 2) and (β + 2).

Case study4 marks

Read the following and answer the questions that follow: During a school sports day, a student throws a ball vertically upwards from ground level. Its height above the ground after t seconds is modelled by the polynomial h(t) = -5t² + 20t, where h is measured in metres and t in seconds. (a) Find the zeroes of h(t) and interpret them in the context of the throw. (b) Verify the relationship between the zeroes and the coefficients of h(t). (c) The ball reaches its greatest height midway between the two zeroes. Find that greatest height. (d) Find the height of the ball 3 seconds after it is thrown.