Assertion (A): The polynomial $x^2 + 4$ has two real zeroes. Reason (R): The graph of $y = x^2 + 4$ intersects the x-axis at two distinct points.

A and R both correct, R explains A

A and R both correct, R does not explain A

Assertion (A): A polynomial of degree $n$ has exactly $n$ zeroes. Reason (R): The graph of a polynomial of degree $n$ intersects the x-axis at most $n$ points.

A and R both correct, R explains A

A and R both correct, R does not explain A

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This quiz covers the definition of polynomials, their degrees, types (linear, quadratic, cubic), finding zeroes, the geometrical interpretation of zeroes, and the relationship between zeroes and coefficients.

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Home › CBSE › Class 10 › Maths › POLYNOMIALS

CBSE · Class 10 · 🧮 Maths · Chapter 2

POLYNOMIALS

Degree of a polynomialLinear, Quadratic, and Cubic PolynomialsZeroes of a polynomialGeometrical meaning of zeroesRelationship between zeroes and coefficients of quadratic polynomialsRelationship between zeroes and coefficients of cubic polynomials

The chapter 'Polynomials' introduces students to different types of polynomials (linear, quadratic, cubic) and their degrees. It delves into the geometrical meaning of the zeroes of a polynomial, explaining how they relate to the x-intercepts of the graph. A crucial part of this chapter is understanding the relationship between the zeroes and the coefficients of quadratic and cubic polynomials, including formulas for sum and product of zeroes. This knowledge is fundamental for further studies in algebra and calculus.

Polynomials ka Introduction aur Types

Class 9 mein humne polynomials ke baare mein padha tha. Ab usko revise karte hain aur kuch naye concepts seekhte hain.

Polynomial: Ek algebraic expression jisme variables ki power hamesha non-negative integer hoti hai.
Example: $4x + 2$, $2y^2 - 3y + 4$, $5x^3 - 4x^2 + x - \sqrt{2}$
Non-polynomials: $\frac{1}{x-1}$, $\sqrt{x} + 2$ (kyunki power fraction hai), $\frac{1}{x^2 + 2x + 3}$ (kyunki power negative ho jayegi)

Degree of a Polynomial: Variable ki highest power polynomial mein.
$4x + 2$ ki degree 1 hai.
$2y^2 - 3y + 4$ ki degree 2 hai.
$5x^3 - 4x^2 + x - \sqrt{2}$ ki degree 3 hai.

Types of Polynomials (Degree ke basis par):
Linear Polynomial: Degree 1 hoti hai.
General form: $ax + b$, jahan $a \neq 0$.
Examples: $2x - 3$, $\sqrt{3}x + 5$, $y + \sqrt{2}$
Quadratic Polynomial: Degree 2 hoti hai.
General form: $ax^2 + bx + c$, jahan $a \neq 0$.
'Quadratic' word 'quadrate' se aaya hai, jiska matlab 'square' hota hai.
Examples: $2x^2 + 3x - \frac{2}{5}$, $y^2 - 2$, $2 - x^2 + \sqrt{3}x$
Cubic Polynomial: Degree 3 hoti hai.
General form: $ax^3 + bx^2 + cx + d$, jahan $a \neq 0$.
Examples: $2 - x^3$, $x^3$, $\sqrt{2}x^3$, $3x^3 - 2x^2 + x - 1$

Value of a Polynomial at a given point:
Agar $p(x)$ ek polynomial hai aur $k$ koi real number hai, toh $x$ ko $k$ se replace karne par jo value milti hai, use $p(x)$ ki value at $x = k$ kehte hain, aur ise $p(k)$ se denote karte hain.
Example: $p(x) = x^2 - 3x - 4$ ke liye,
$p(2) = 2^2 - 3(2) - 4 = 4 - 6 - 4 = -6$
$p(-1) = (-1)^2 - 3(-1) - 4 = 1 + 3 - 4 = 0$

Zero of a Polynomial:
Ek real number $k$ ko polynomial $p(x)$ ka zero kehte hain, agar $p(k) = 0$ ho.
Upar wale example mein, $-1$ aur $4$ polynomial $x^2 - 3x - 4$ ke zeroes hain, kyunki $p(-1) = 0$ aur $p(4) = 0$.
Linear polynomial $ax + b$ ka zero hamesha $-\frac{b}{a}$ hota hai. (i.e., $-\frac{\text{Constant term}}{\text{Coefficient of x}}$)
Example: $p(x) = 2x + 3$ ka zero $-\frac{3}{2}$ hai.

❗Important

Polynomials mein variable ki power hamesha whole number honi chahiye. Fraction ya negative power wale expressions polynomials nahi hote.

📖Definition

Zero of a Polynomial: Ek real number $k$ polynomial $p(x)$ ka zero tab hota hai jab $p(k) = 0$.

Zeroes ka Geometrical Meaning

Zeroes ka matlab kya hota hai graph par? Ye samajhna bahut important hai.

Linear Polynomial (Degree 1):
$y = ax + b$ ka graph hamesha ek straight line hota hai.
Iska zero woh $x$-coordinate hota hai jahan graph $x$-axis ko cut karta hai.
Ek linear polynomial ka exactly ek zero hota hai, jo $-\frac{b}{a}$ hota hai.
Example: $y = 2x + 3$ ka graph $x$-axis ko $(-\frac{3}{2}, 0)$ par cut karta hai. So, zero is $-\frac{3}{2}$.

Quadratic Polynomial (Degree 2):
$y = ax^2 + bx + c$ ka graph hamesha ek parabola hota hai.
Parabola ki shape $a$ ke sign par depend karti hai:
Agar $a > 0$ hai, toh parabola upwards open hota hai ($U$ shape).
Agar $a < 0$ hai, toh parabola downwards open hota hai ($\cap$ shape).
Quadratic polynomial ke zeroes woh $x$-coordinates hote hain jahan graph $x$-axis ko cut karta hai.
Ek quadratic polynomial ke at most 2 zeroes ho sakte hain. Iske teen cases possible hain:
Case 1: Two distinct zeroes: Graph $x$-axis ko do alag-alag points par cut karta hai. (e.g., $y = x^2 - 3x - 4$ ke zeroes $-1$ aur $4$ hain)
Case 2: One zero (two equal zeroes): Graph $x$-axis ko ek point par touch karta hai (tangent hota hai).
Case 3: No zero: Graph $x$-axis ko bilkul cut nahi karta (ya toh poora upar hota hai ya poora neeche).

Cubic Polynomial (Degree 3):
$y = ax^3 + bx^2 + cx + d$ ka graph ek curve hota hai.
Cubic polynomial ke zeroes woh $x$-coordinates hote hain jahan graph $x$-axis ko cut karta hai.
Ek cubic polynomial ke at most 3 zeroes ho sakte hain.
Example: $y = x^3 - 4x$ ke zeroes $-2, 0, 2$ hain.

General Rule:
Ek polynomial $p(x)$ jiski degree $n$ hai, uske at most $n$ zeroes ho sakte hain.
Zeroes ki sankhya graph ke $x$-axis ko intersect karne wale points ki sankhya ke barabar hoti hai.

💡Tip

Graph dekh kar zeroes count karna board exams mein common question hai. Sirf $x$-axis par intersection points count karo.

Zeroes aur Coefficients ke beech Relationship

Ye chapter ka sabse important section hai. Zeroes aur coefficients ke beech ke relationships ko yaad rakhna aur apply karna aana chahiye.

Linear Polynomial:
$p(x) = ax + b$
Zero: $k = -\frac{b}{a}$
Relationship: Zero \(= -\frac{\text{Constant term}}{\text{Coefficient of x}}

Quadratic Polynomial:
$p(x) = ax^2 + bx + c$, jahan $a \neq 0$.
Agar $\alpha$ aur $\beta$ iske zeroes hain, toh:
Sum of Zeroes: $\alpha + \beta = -\frac{b}{a} = -\frac{\text{Coefficient of x}}{\text{Coefficient of x}^2}$
Product of Zeroes: $\alpha \beta = \frac{c}{a} = \frac{\text{Constant term}}{\text{Coefficient of x}^2}$
Polynomial banana (given zeroes or sum/product):
Agar zeroes $\alpha$ aur $\beta$ diye hain, toh quadratic polynomial $k[x^2 - (\alpha + \beta)x + \alpha \beta]$ hoga, jahan $k$ koi non-zero constant hai.
Ya phir, $x^2 - (\text{Sum of Zeroes})x + (\text{Product of Zeroes})$

Cubic Polynomial:
$p(x) = ax^3 + bx^2 + cx + d$, jahan $a \neq 0$.
Agar $\alpha, \beta, \gamma$ iske zeroes hain, toh:
Sum of Zeroes: $\alpha + \beta + \gamma = -\frac{b}{a} = -\frac{\text{Coefficient of x}^2}{\text{Coefficient of x}^3}$
Sum of the products of zeroes taken two at a time: $\alpha \beta + \beta \gamma + \gamma \alpha = \frac{c}{a} = \frac{\text{Coefficient of x}}{\text{Coefficient of x}^3}$
Product of Zeroes: $\alpha \beta \gamma = -\frac{d}{a} = -\frac{\text{Constant term}}{\text{Coefficient of x}^3}$

Verification Steps:

Polynomial ke zeroes find karo (splitting the middle term, quadratic formula, ya hit & trial se).
Zeroes ka sum aur product calculate karo.
Formula $-\frac{b}{a}$ aur $\frac{c}{a}$ se sum aur product calculate karo.
Dono results ko compare karo. Agar same hain, toh verified.

🧮Formula

Quadratic Polynomial ($ax^2 + bx + c$)

Sum of zeroes ($\alpha + \beta$) $= -\frac{b}{a}$
Product of zeroes ($\alpha \beta$) $= \frac{c}{a}$

Cubic Polynomial ($ax^3 + bx^2 + cx + d$)

Sum of zeroes ($\alpha + \beta + \gamma$) $= -\frac{b}{a}$
Sum of product of zeroes taken two at a time ($\alpha \beta + \beta \gamma + \gamma \alpha$) $= \frac{c}{a}$
Product of zeroes ($\alpha \beta \gamma$) $= -\frac{d}{a}$

🚧Misconception

Signs ka dhyan rakho! Sum of zeroes mein $-\frac{b}{a}$ hota hai, aur cubic ke product mein $-\frac{d}{a}$. Ye negative signs bhoolna common mistake hai.

Division Algorithm for Polynomials

Jaise numbers ke liye division algorithm hota hai (Dividend = Divisor $\times$ Quotient + Remainder), waise hi polynomials ke liye bhi hota hai.

Division Algorithm Statement:
Agar $p(x)$ aur $g(x)$ koi do polynomials hain, jahan $g(x) \neq 0$, toh hum $q(x)$ (quotient) aur $r(x)$ (remainder) aise polynomials find kar sakte hain ki:

$p(x) = g(x) \times q(x) + r(x)$

Jahan $r(x) = 0$ ya degree of $r(x)$ < degree of $g(x)$ ho.

Steps for Polynomial Division (Long Division Method):

Arrange: Both polynomials $p(x)$ aur $g(x)$ ko descending powers of variable mein arrange karo.
Divide first terms: $p(x)$ ke first term ko $g(x)$ ke first term se divide karo. Ye quotient ka first term hoga.
Multiply: Quotient ke first term ko poore $g(x)$ se multiply karo.
Subtract: Result ko $p(x)$ se subtract karo. Signs change karna mat bhoolna.
Bring down: Next term ko neeche lao.
Repeat: Steps 2-5 repeat karo jab tak remainder ki degree divisor ki degree se kam na ho jaye ya remainder zero na ho jaye.

Important Points:
Agar $r(x) = 0$ hai, toh $g(x)$ (aur $q(x)$ bhi) $p(x)$ ka factor hota hai.
Agar kisi polynomial ke zeroes diye hain, toh unse factors bana kar long division se baaki zeroes find kar sakte hain.
Example: Agar $\sqrt{2}$ aur $-\sqrt{2}$ polynomial $p(x)$ ke zeroes hain, toh $(x - \sqrt{2})$ aur $(x + \sqrt{2})$ factors hain. Inka product $(x^2 - 2)$ bhi $p(x)$ ka factor hoga. Phir $p(x)$ ko $(x^2 - 2)$ se divide karke quotient ke zeroes find kar sakte hain.

⭐Remember

Division Algorithm: $p(x) = g(x) \times q(x) + r(x)$. Is formula ko hamesha yaad rakho. Verification mein kaam aata hai.

💡Tip

Long division mein signs change karna sabse common mistake hai. Jab subtract karte ho, toh har term ka sign ulta ho jata hai.

Easy (15)

What is the degree of the polynomial $p(x) = 5x^3 - 4x^2 + x - \sqrt{2}$?
- 1
- 2
- 3 (correct)
- 4
Why: The degree is the highest power of the variable in the polynomial.
A polynomial of degree 1 is called a _________ polynomial.
Why: A polynomial with degree 1 is a linear polynomial.
The expression $1/(x-1)$ is a polynomial.
Why: Polynomials cannot have variables in the denominator.
What is the value of the polynomial $p(x) = x^2 - 3x - 4$ at $x = 0$?
- 0
- 1
- -4 (correct)
- 4
Why: Substitute x=0 into the polynomial to find its value.
A real number $k$ is a zero of a polynomial $p(x)$ if $p(k)$ is equal to _________.
Why: A zero of a polynomial makes the polynomial's value zero.
The graph of a linear polynomial $y = ax + b$ is always a straight line.
Why: Linear polynomials graph as straight lines.
The zeroes of a polynomial are the x-coordinates where its graph intersects the _________.
Why: Zeroes correspond to where the graph crosses the x-axis.
A quadratic polynomial can have at most how many zeroes?
- 1
- 2 (correct)
- 3
- 4
Why: The maximum number of zeroes for a polynomial is its degree.
The graph of a quadratic polynomial $y = ax^2 + bx + c$ is called a parabola.
Why: The graph of a quadratic polynomial is always a parabola.
If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $ax^2 + bx + c$, then their sum $\alpha + \beta$ is equal to $b/a$.
Why: The sum of zeroes is -b/a, not b/a.
The zero of the linear polynomial $ax + b$ is given by the formula _________.
Why: Set the polynomial to zero and solve for x.
Which of the following is a quadratic polynomial?
- $2x - 3$
- $x^3 + 1$
- $2x^2 + 3x - 2/5$ (correct)
- $1/x$
Why: A quadratic polynomial has a degree of 2.
If $p(x) = x^2 - 3x - 4$, then $p(4) = 0$.
Why: Substitute x=4 into the polynomial to verify.
The product of the zeroes of a quadratic polynomial $ax^2 + bx + c$ is given by _________.
Why: Recall the formula for the product of zeroes.
A polynomial of degree 3 is called a cubic polynomial.
Why: This is the definition of a cubic polynomial.

Medium (15)

Find the zeroes of the quadratic polynomial $x^2 + 7x + 10$.
- 2, 5
- -2, -5 (correct)
- -2, 5
- 2, -5
Why: Factorize the polynomial to find its zeroes.
Match the polynomial type with its degree.
Why: Polynomial types are defined by their highest degree.
If the sum of the zeroes of a quadratic polynomial is -3 and the product is 2, what is one possible quadratic polynomial?
- $x^2 - 3x + 2$
- $x^2 + 3x - 2$
- $x^2 + 3x + 2$ (correct)
- $x^2 - 2x + 3$
Why: Use the relationship $x^2 - (\alpha + \beta)x + \alpha\beta = 0$.
The geometrical representation of a quadratic polynomial is a curve called a _________.
Why: This specific curve shape is known as a parabola.
Arrange the steps to find the zeroes of $2x^2 - 8x + 6$ by splitting the middle term.
Why: Follow the standard procedure for splitting the middle term.
For the quadratic polynomial $x^2 - 3$, what is the sum of its zeroes?
- $\sqrt{3}$
- $-\sqrt{3}$
- 0 (correct)
- 3
Why: Use the formula $\alpha + \beta = -b/a$.
If the graph of a quadratic polynomial intersects the x-axis at exactly one point, it means the polynomial has _________ equal zeroes.
Why: One intersection point implies two coincident zeroes.
Match the polynomial with its degree.
Why: The degree is the highest power of the variable.
What is the product of the zeroes of the polynomial $3x^2 + 5x - 2$?
- $5/3$
- $-5/3$
- $2/3$
- $-2/3$ (correct)
Why: Use the formula $\alpha\beta = c/a$.
Order the following polynomial types by their maximum number of zeroes, from least to greatest.
Why: The maximum number of zeroes equals the degree of the polynomial.
The graph of $y = x^2 - 3x - 4$ intersects the x-axis at which points?
- (0, -4) and (1, -6)
- (-1, 0) and (4, 0) (correct)
- (-2, 6) and (5, 6)
- (0, 0) and (3, 0)
Why: The x-intercepts are the zeroes of the polynomial.
Match the polynomial type with its general form.
Why: Recall the standard forms for each polynomial type.
If the graph of $y = p(x)$ does not intersect the x-axis at any point, then the quadratic polynomial $p(x)$ has _________ real zeroes.
Why: No x-intercepts means no real zeroes.
Which of the following describes the graph of $y = ax^2 + bx + c$ when $a < 0$?
- Opens upwards (∪)
- Opens downwards (∩) (correct)
- A straight line
- A cubic curve
Why: The sign of 'a' determines the opening direction of a parabola.
Arrange the terms of the cubic polynomial $ax^3 + bx^2 + cx + d$ in descending order of their degrees.
Why: Standard form lists terms from highest to lowest degree.

Hard (10)

If the zeroes of the quadratic polynomial $x^2 + (a+1)x + b$ are 2 and -3, then find the values of $a$ and $b$.
Why: Use sum and product of zeroes formulas to form equations for a and b.
Assertion (A): The polynomial $x^2 + 4$ has two real zeroes. Reason (R): The graph of $y = x^2 + 4$ intersects the x-axis at two distinct points.
- A and R both correct, R explains A
- A and R both correct, R does not explain A
- A correct, R wrong
- A wrong, R correct
- Both A and R are wrong (correct)
Why: The polynomial $x^2+4$ has no real zeroes, and its graph does not intersect the x-axis.
A rectangular garden has an area represented by the polynomial $x^2 - 5x - 14$. If the length of the garden is $(x-7)$ units, what is the width of the garden?
- $(x-2)$ units
- $(x+2)$ units (correct)
- $(x-14)$ units
- $(x+7)$ units
Why: Divide the area polynomial by the length polynomial to find the width.
The graph of a polynomial $y = p(x)$ is shown below. How many zeroes does $p(x)$ have?
- 1
- 2
- 3
- 4 (correct)
Why: Count the number of times the graph intersects the x-axis.
If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $p(x) = x^2 - 5x + k$ such that $\alpha - \beta = 1$, find the value of $k$.
Why: Use sum and product of zeroes along with the given difference.
Assertion (A): A polynomial of degree $n$ has exactly $n$ zeroes. Reason (R): The graph of a polynomial of degree $n$ intersects the x-axis at most $n$ points.
- A and R both correct, R explains A
- A and R both correct, R does not explain A
- A correct, R wrong
- A wrong, R correct (correct)
- Both A and R are wrong
Why: A polynomial of degree n has AT MOST n zeroes, not exactly n. The reason is correct.
A company's profit (in lakhs) for the last quarter can be modeled by the polynomial $P(x) = x^3 - 4x^2 + 3x$, where $x$ is the number of units sold (in thousands). If the company breaks even (profit is zero) at $x=1$ thousand units, what are the other two break-even points?
- x=2, x=3
- x=0, x=3 (correct)
- x=0, x=4
- x=1, x=4
Why: Factorize the polynomial and find its zeroes.
The graph of $y = x^3 - 4x$ is shown below. What are its zeroes?
- -1, 0, 1
- -2, 0, 2 (correct)
- 0, 1, 2
- -2, -1, 0
Why: Identify the x-intercepts from the graph.
If $\alpha, \beta, \gamma$ are the zeroes of the cubic polynomial $x^3 - 3x^2 - x + 3$, find the value of $\alpha\beta + \beta\gamma + \gamma\alpha$.
Why: Use the formula $\alpha\beta + \beta\gamma + \gamma\alpha = c/a$ for cubic polynomials.
Find the value of $k$ if $x+1$ is a factor of the polynomial $2x^2 + kx - 1$.
Why: If x+1 is a factor, then x=-1 is a zero of the polynomial.