POLYNOMIALS - Q&A
EXERCISE 2.11. The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.
(Note: Based on standard NCERT graphs for this exercise)
(i) Graph is a straight line parallel to the x-axis.
(ii) Graph intersects the x-axis at one point.
(iii) Graph intersects the x-axis at three points.
(iv) Graph intersects the x-axis at two points.
(v) Graph intersects the x-axis at four points.
(vi) Graph intersects the x-axis at three points (touching at two, cutting at one).
The number of zeroes of a polynomial p(x) is equal to the number of points where the graph of y = p(x) intersects the x-axis.
(i) The graph does not intersect the x-axis at any point.
Answer: 0
(ii) The graph intersects the x-axis at exactly one point.
Answer: 1
(iii) The graph intersects the x-axis at three points.
Answer: 3
(iv) The graph intersects the x-axis at two points.
Answer: 2
(v) The graph intersects the x-axis at four points.
Answer: 4
(vi) The graph intersects the x-axis at three points.
Answer: 3
EXERCISE 2.2
1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) x2 - 2x - 8
Solution:
Step 1: Find the zeroes by factorization.
We need to split the middle term (-2x) such that the sum is -2x and the product is -8x2.
x2 - 4x + 2x - 8 = 0
x(x - 4) + 2(x - 4) = 0
(x - 4)(x + 2) = 0
So, x - 4 = 0 or x + 2 = 0
x = 4 or x = -2
The zeroes are 4 and -2.
Step 2: Verify the relationship.
Here, a = 1, b = -2, c = -8.
Sum of zeroes = 4 + (-2) = 2
-Coefficient of x / Coefficient of x2 = -b/a = -(-2)/1 = 2
Sum of zeroes = -b/a (Verified)
Product of zeroes = 4 × (-2) = -8
Constant term / Coefficient of x2 = c/a = -8/1 = -8
Product of zeroes = c/a (Verified)
(ii) 4s2 - 4s + 1
Solution:
Step 1: Find the zeroes.
4s2 - 2s - 2s + 1 = 0
2s(2s - 1) - 1(2s - 1) = 0
(2s - 1)(2s - 1) = 0
So, s = 1/2, 1/2
The zeroes are 1/2 and 1/2.
Step 2: Verify the relationship.
Here, a = 4, b = -4, c = 1.
Sum of zeroes = 1/2 + 1/2 = 1
-b/a = -(-4)/4 = 4/4 = 1
(Verified)
Product of zeroes = 1/2 × 1/2 = 1/4
c/a = 1/4
(Verified)
(iii) 6x2 - 3 - 7x
Solution:
Step 1: Rearrange and find zeroes.
Rearrange in standard form: 6x2 - 7x - 3
Split middle term (product -18, sum -7):
6x2 - 9x + 2x - 3 = 0
3x(2x - 3) + 1(2x - 3) = 0
(3x + 1)(2x - 3) = 0
x = -1/3 or x = 3/2
The zeroes are -1/3 and 3/2.
Step 2: Verify the relationship.
Here, a = 6, b = -7, c = -3.
Sum of zeroes = -1/3 + 3/2 = (-2 + 9)/6 = 7/6
-b/a = -(-7)/6 = 7/6
(Verified)
Product of zeroes = (-1/3) × (3/2) = -3/6 = -1/2
c/a = -3/6 = -1/2
(Verified)
(iv) 4u2 + 8u
Solution:
Step 1: Find zeroes.
4u(u + 2) = 0
u = 0 or u + 2 = 0
The zeroes are 0 and -2.
Step 2: Verify relationship.
Here, a = 4, b = 8, c = 0.
Sum of zeroes = 0 + (-2) = -2
-b/a = -8/4 = -2
(Verified)
Product of zeroes = 0 × -2 = 0
c/a = 0/4 = 0
(Verified)
(v) t2 - 15
Solution:
Step 1: Find zeroes.
t2 = 15
t = √15 or t = -√15
The zeroes are √15 and -√15.
Step 2: Verify relationship.
Here, a = 1, b = 0 (no 't' term), c = -15.
Sum of zeroes = √15 - √15 = 0
-b/a = -0/1 = 0
(Verified)
Product of zeroes = (√15)( -√15) = -15
c/a = -15/1 = -15
(Verified)
(vi) 3x2 - x - 4
Solution:
Step 1: Find zeroes.
Split middle term (product -12, sum -1):
3x2 - 4x + 3x - 4 = 0
x(3x - 4) + 1(3x - 4) = 0
(x + 1)(3x - 4) = 0
x = -1 or x = 4/3
The zeroes are -1 and 4/3.
Step 2: Verify relationship.
Here, a = 3, b = -1, c = -4.
Sum of zeroes = -1 + 4/3 = (-3 + 4)/3 = 1/3
-b/a = -(-1)/3 = 1/3
(Verified)
Product of zeroes = -1 × 4/3 = -4/3
c/a = -4/3
(Verified)
2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
(Note: A quadratic polynomial is given by k[x2 - (Sum of zeroes)x + (Product of zeroes)], where k is a constant).
(i) 1/4, -1
Solution:
Sum (S) = 1/4, Product (P) = -1
Polynomial = x2 - Sx + P
= x2 - (1/4)x + (-1)
= x2 - x/4 - 1
To remove the fraction, we can multiply by 4 (taking k=4).
4x2 - x - 4
(ii) √2, 1/3
Solution:
Sum (S) = √2, Product (P) = 1/3
Polynomial = x2 - √2x + 1/3
To remove the fraction, multiply by 3.
3x2 - 3√2x + 1
(iii) 0, √5
Solution:
Sum (S) = 0, Product (P) = √5
Polynomial = x2 - (0)x + √5
x2 + √5
(iv) 1, 1
Solution:
Sum (S) = 1, Product (P) = 1
Polynomial = x2 - 1x + 1
x2 - x + 1
(v) -1/4, 1/4
Solution:
Sum (S) = -1/4, Product (P) = 1/4
Polynomial = x2 - (-1/4)x + 1/4
= x2 + x/4 + 1/4
Multiplying by 4:
4x2 + x + 1
(vi) 4, 1
Solution:
Sum (S) = 4, Product (P) = 1
Polynomial = x2 - 4x + 1
x2 - 4x + 1
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Quick Review Flashcards - Click to flip and test your knowledge!
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The highest power of x in p(x).
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1
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The powers of the variable must be non-negative integers.
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ax + b, where a \ne 0.
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It is derived from 'quadrate', which means 'square'.
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ax^3 + bx^2 + cx + d, where a, b, c, d are real numbers and a \ne 0.
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The value of p(x) obtained by replacing x with k.
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When p(k) = 0.
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-1 and 4.
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Answer
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Geometrically, what do the zeroes of a polynomial p(x) represent on a graph?
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The x-coordinates of the points where the graph of y = p(x) intersects the x-axis.
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A straight line.
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Answer
One point.
Question
The graph of y = ax^2 + bx + c is a curve called a _____.
Answer
Parabola
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When a > 0.
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Under what condition does the parabola y = ax^2 + bx + c open downwards?
Answer
When a < 0.
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If a quadratic parabola intersects the x-axis at two distinct points A and A', how many zeroes does the polynomial have?
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Two distinct zeroes.
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The two zeroes are equal (or coincident).
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Zero (it has no zeroes).
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At most 2 zeroes.
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At most 3 zeroes.
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Answer
n
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Answer
By counting the number of points where the graph intersects the x-axis.
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\alpha + \beta = -\frac{b}{a}
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Answer
a^2 - b^2 = (a - b)(a + b)
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Find the zeroes of the polynomial x^2 - 3.
Answer
\sqrt{3} and -\sqrt{3}.
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What is the quadratic polynomial where the sum of zeroes is -3 and the product of zeroes is 2?
Answer
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For a cubic polynomial ax^3 + bx^2 + cx + d, what is the formula for the sum of zeroes \alpha + \beta + \gamma?
Answer
-\frac{b}{a}
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For a cubic polynomial ax^3 + bx^2 + cx + d, what is the formula for the product of zeroes \alpha \beta \gamma?
Answer
-\frac{d}{a}
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For a cubic polynomial ax^3 + bx^2 + cx + d, what is the formula for \alpha \beta + \beta \gamma + \gamma \alpha?
Answer
\frac{c}{a}
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Verify if 3, -1, and -\frac{1}{3} are zeroes of p(x) = 3x^3 - 5x^2 - 11x - 3.
Answer
Yes, they are zeroes as p(3) = 0, p(-1) = 0, and p(-\frac{1}{3}) = 0.
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In a cubic polynomial, the sum of the products of the zeroes taken two at a time equals \frac{\text{coefficient of } x}{\text{?}}.
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Answer
a=3, b=-5, c=-11, d=-3
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Answer
Parabola
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-4 (calculated as -\frac{8}{2}).
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0 and 1.
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A polynomial of degree 1 is called a _____ polynomial.
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a \ne 0
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What is the value of the constant term d in the cubic polynomial 3x^3 - 5x^2 - 11x - 3?
Answer
-3