Question 1

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.

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Accepted Answer

To find the number of zeroes of each polynomial from the graphs given in Fig. 2.10, observe the points where the graph intersects the x-axis. Each intersection corresponds to a zero of the polynomial.

(i) Count the number of x-intercepts in graph (i).
(ii) Count the number of x-intercepts in graph (ii).
(iii) Count the number of x-intercepts in graph (iii).
(iv) Count the number of x-intercepts in graph (iv).
(v) Count the number of x-intercepts in graph (v).
(vi) Count the number of x-intercepts in graph (vi).

The exact number depends on the graphs shown in Fig. 2.10, which are not reproduced here. Generally, the number of zeroes equals the number of distinct points where the graph crosses or touches the x-axis. — The zeroes of a polynomial correspond to the values of x for which y = p(x) = 0. Graphically, these are the points where the curve intersects the x-axis. By counting these points on each graph, we determine the number of zeroes of the polynomial.

Question 2

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(i) x^{2} - 2x - 8

(ii) 4s^{2} - 4s + 1

(iii) 6x^{2} - 3 - 7x

(iv) 4u^{2} + 8u

(v) t^2 - 15

(vi) 3x^{2} - x - 4

Accepted Answer

Solution:

(i) Polynomial: x^2 - 2x - 8
Sum of zeroes, α + β = -b/a = -(-2)/1 = 2
Product of zeroes, αβ = c/a = -8/1 = -8

Find zeroes by factorization:
 x^2 - 2x - 8 = (x - 4)(x + 2)
Zeroes are 4 and -2
Sum = 4 + (-2) = 2, Product = 4 * (-2) = -8
Relationship verified.

(ii) Polynomial: 4s^2 - 4s + 1
Sum = -b/a = -(-4)/4 = 1
Product = c/a = 1/4

Find zeroes using quadratic formula:
s = [4 ± sqrt(16 - 16)] / (2*4) = 4/(8) = 0.5
Since discriminant is zero, both zeroes are equal: 0.5 and 0.5
Sum = 0.5 + 0.5 = 1, Product = 0.5 * 0.5 = 0.25 = 1/4
Relationship verified.

(iii) Polynomial: 6x^2 - 3 - 7x
Rewrite as 6x^2 - 7x - 3
Sum = -b/a = -(-7)/6 = 7/6
Product = c/a = -3/6 = -1/2

Find zeroes using quadratic formula:
x = [7 ± sqrt(49 + 72)] / 12 = [7 ± sqrt(121)] / 12
= [7 ± 11] / 12
Zeroes:
 x1 = (7 + 11)/12 = 18/12 = 3/2
 x2 = (7 - 11)/12 = -4/12 = -1/3
Sum = 3/2 + (-1/3) = (9/6 - 2/6) = 7/6
Product = (3/2)*(-1/3) = -1/2
Relationship verified.

(iv) Polynomial: 4u^2 + 8u
Sum = -b/a = -8/4 = -2
Product = c/a = 0/4 = 0

Find zeroes by factorization:
4u^2 + 8u = 4u(u + 2)
Zeroes are 0 and -2
Sum = 0 + (-2) = -2
Product = 0 * (-2) = 0
Relationship verified.

(v) Polynomial: t^2 - 15
Sum = -b/a = 0/1 = 0
Product = c/a = -15/1 = -15

Zeroes are ±√15
Sum = √15 + (-√15) = 0
Product = (√15)(-√15) = -15
Relationship verified.

(vi) Polynomial: 3x^2 - x - 4
Sum = -b/a = -(-1)/3 = 1/3
Product = c/a = -4/3

Find zeroes using quadratic formula:
x = [1 ± sqrt(1 + 48)] / 6 = [1 ± 7] / 6
Zeroes:
 x1 = (1 + 7)/6 = 8/6 = 4/3
 x2 = (1 - 7)/6 = -6/6 = -1
Sum = 4/3 + (-1) = 1/3
Product = (4/3)*(-1) = -4/3
Relationship verified. — For each quadratic polynomial ax^2 + bx + c, the sum of zeroes α + β = -b/a and product αβ = c/a.

We find zeroes either by factorization or quadratic formula, then verify these relationships by calculating sum and product of zeroes and comparing with coefficients.

Question 3

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

(i) \frac{1}{4}, -1

(ii) \sqrt{2}, \frac{1}{3}

(iii) 0, \sqrt{5}

(iv) 1, 1

(v) -\frac{1}{4}, \frac{1}{4}

(vi) 4, 1

Accepted Answer

Solution:

Recall: For quadratic polynomial with zeroes α and β,
Sum = α + β = -b/a and Product = αβ = c/a.

We take a = 1 for simplicity, so polynomial is x^2 - (sum)x + product.

(i) Sum = 1/4, Product = -1
Polynomial: x^2 - (1/4)x - 1

(ii) Sum = √2, Product = 1/3
Polynomial: x^2 - (\sqrt{2})x + 1/3

(iii) Sum = 0, Product = √5
Polynomial: x^2 - 0*x + \sqrt{5} = x^2 + \sqrt{5}

(iv) Sum = 1, Product = 1
Polynomial: x^2 - x + 1

(v) Sum = -1/4, Product = 1/4
Polynomial: x^2 - (-1/4)x + 1/4 = x^2 + (1/4)x + 1/4

(vi) Sum = 4, Product = 1
Polynomial: x^2 - 4x + 1 — Using the relation for quadratic polynomials with zeroes α and β, the polynomial can be constructed as x^2 - (sum of zeroes)x + (product of zeroes).

Substitute the given sum and product values to get the required polynomial.

Question 4

Which of the following expressions is NOT a polynomial?

Accepted Answer

\sqrt{x} + 2 — Polynomials are algebraic expressions involving variables raised to non-negative integer powers with real coefficients. Expressions like \sqrt{x} + 2 involve fractional powers (square root), which are not allowed in polynomials. Options A, B, and D are polynomials with degrees 1, 2, and 3 respectively.

Question 5

What is the degree of the polynomial $7u^6 - \frac{3}{2} u^3 + 4u^2 + u - 8$?

Accepted Answer

6 — The degree of a polynomial is the highest power of the variable present. Here, the highest power of $u$ is 6 in the term $7u^6$, so the degree is 6.

Question 6

Identify the correct general form of a quadratic polynomial in $x$.

Accepted Answer

$ax^2 + bx + c$, where $a 
eq 0$ — A quadratic polynomial in $x$ has degree 2 and is generally written as $ax^2 + bx + c$ with $a 
eq 0$. Option A is linear, C is cubic, and D is a cubic polynomial but not the standard form for quadratic.

Question 7

If $p(x) = x^2 - 3x - 4$, what is the value of $p(2)$?

Accepted Answer

-6 — Given: p(x) = x^2 - 3x - 4, x = 2
Find: p(2)
Formula: p(k) = k^2 - 3k - 4
Solution:
Step 1: Substitute x = 2
Step 2: p(2) = 2^2 - 3 × 2 - 4
Step 3: p(2) = 4 - 6 - 4 = -6
Answer: -6
Note: Ensure correct order of operations and signs.

Question 8

For the polynomial $p(x) = x^2 - 3x - 4$, which of the following are its zeroes?

Accepted Answer

-1 and 4 — Zeroes of a polynomial are values of $x$ for which $p(x) = 0$. Given $p(-1) = (-1)^2 - 3(-1) - 4 = 1 + 3 - 4 = 0$ and $p(4) = 16 - 12 - 4 = 0$, so zeroes are -1 and 4.

Question 9

What is the zero of the linear polynomial $p(x) = 2x + 3$?

Accepted Answer

-\frac{3}{2} — Zero of linear polynomial $ax + b$ is $-\frac{b}{a}$. Here, $a=2$, $b=3$, so zero is $-\frac{3}{2}$.

Question 10

The graph of $y = 2x + 3$ intersects the x-axis at which point?

Accepted Answer

\left(-\frac{3}{2}, 0\right) — The x-intercept occurs where $y=0$. Solve $0 = 2x + 3$ gives $x = -\frac{3}{2}$. So the point is $\left(-\frac{3}{2}, 0\right)$.

Question 11

Which of the following statements about the graph of a quadratic polynomial $y = ax^2 + bx + c$ is TRUE?

Accepted Answer

The graph is a parabola opening upwards if $a > 0$ and downwards if $a < 0$. — The graph of $y = ax^2 + bx + c$ is a parabola. It opens upwards if $a > 0$ and downwards if $a < 0$. It may intersect the x-axis at zero, one, or two points depending on the zeroes.

Question 12

From the table below for $y = x^2 - 3x - 4$, which values of $x$ correspond to zeroes of the polynomial?

| x | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
|---|----|----|---|---|---|---|---|---|
| y | 6  | 0  | -4| -6| -6| -4| 0 | 6 |

Accepted Answer

-1 and 4 — Zeroes are values of $x$ where $y=0$. From the table, $y=0$ at $x=-1$ and $x=4$, so these are the zeroes.

Question 13

Which of the following cases is NOT possible for the graph of a quadratic polynomial $y = ax^2 + bx + c$?

Accepted Answer

The graph intersects the x-axis at three points. — A quadratic polynomial graph is a parabola and can intersect the x-axis at zero, one, or two points only. Three intersections are not possible.

Question 14

Consider the cubic polynomial $p(x) = x^3 - 4x$. Which of the following are its zeroes?

Accepted Answer

-2, 0, 2 — Checking $p(-2) = (-2)^3 - 4(-2) = -8 + 8 = 0$, $p(0) = 0 - 0 = 0$, $p(2) = 8 - 8 = 0$. So, zeroes are -2, 0, and 2.

Question 15

How many zeroes can a cubic polynomial have at most?

Accepted Answer

3 — A polynomial of degree $n$ can have at most $n$ zeroes. For cubic polynomials (degree 3), the maximum number of zeroes is 3.

Mathematics

Mathematics — Study Notes

2.1 Introduction

2.2 Geometrical Meaning of the Zeroes of a Polynomial

Exercise 2.1

Practice Questions — Mathematics

All 14 Chapters in Mathematics