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THE RISE OF NATIONALISM IN EUROPE

Class 10 . Social Science . India And Contemporary World II . The Rise of Nationalism in Europe

Chapter 1: THE RISE OF NATIONALISM IN EUROPE

Ch 1

Social Science

Class 10

Prepzy Class 10 · Mathematics
Study Notes · Maths
Mathematics · Class 10

Polynomials

2 lectures · Quick-revision notes · tap the buttons to check yourself

Lecture 1

Introduction

Reference video · Lecture 1Introduction

Polynomials, Degree and Types

Values and Zeroes of a Polynomial

Worked example

Evaluate $p(x)=x^2-3x-4$ at $x=2,0,-1,4$ and identify its zeroes.

  1. Substitute $x=2$.
    $$p(2)=2^2-3(2)-4=4-6-4=-6$$
  2. Substitute $x=0$.
    $$p(0)=0^2-3(0)-4=-4$$
  3. Substitute $x=-1$.
    $$p(-1)=(-1)^2-3(-1)-4=1+3-4=0$$
  4. Substitute $x=4$.
    $$p(4)=4^2-3(4)-4=16-12-4=0$$
Answer: $p(2)=-6$, $p(0)=-4$, $p(-1)=0$ and $p(4)=0$; the zeroes are $-1$ and $4$.

Geometrical Meaning of Zeroes

Maximum number of zeroes

A polynomial of degree $n$ has at most $n$ zeroes, since its graph can intersect the $x$-axis at at most $n$ points.

The line $y=2x+3$ meets the $x$-axis at $\left(-\frac{3}{2},0\right)$; its zero is $-\frac{3}{2}$.
The line $y=2x+3$ meets the $x$-axis at $\left(-\frac{3}{2},0\right)$; its zero is $-\frac{3}{2}$.
A quadratic graph with its $x$-intercepts.
A quadratic graph with its $x$-intercepts.
Quadratic graphs can touch the $x$-axis at one point.
Quadratic graphs can touch the $x$-axis at one point.
Quadratic graphs may lie entirely above or below the $x$-axis.
Quadratic graphs may lie entirely above or below the $x$-axis.
Worked example

How many zeroes does each of the six polynomial graphs have?

  1. Count the $x$-axis intersections for each graph.
    $$(i)\ 1,\quad (ii)\ 2,\quad (iii)\ 3$$
  2. Continue counting the distinct intersection points.
    $$(iv)\ 1,\quad (v)\ 1,\quad (vi)\ 4$$
Answer: (i) $1$, (ii) $2$, (iii) $3$, (iv) $1$, (v) $1$, and (vi) $4$ zeroes.

Relations Between Zeroes and Coefficients

Quadratic Zeroes and Coefficients
$$\alpha+\beta=-\frac{b}{a},\qquad \alpha\beta=\frac{c}{a}$$

For zeroes $\alpha,\beta$ of $ax^2+bx+c$, where $a
e0$.

Cubic Zeroes and Coefficients
$$\alpha+\beta+\gamma=-\frac{b}{a},\qquad \alpha\beta+\beta\gamma+\gamma\alpha=\frac{c}{a},\qquad \alpha\beta\gamma=-\frac{d}{a}$$

For zeroes $\alpha,\beta,\gamma$ of $ax^3+bx^2+cx+d$, where $a
e0$.

Worked example

Verify the coefficient relations for $x^2-3x-4$.

  1. Read the coefficients and zeroes.
    $$a=1,\quad b=-3,\quad c=-4,\quad \alpha=-1,\quad \beta=4$$
  2. Check the sum relation.
    $$\alpha+\beta=-1+4=3=-\frac{-3}{1}=-\frac{b}{a}$$
  3. Check the product relation.
    $$\alpha\beta=(-1)(4)=-4=\frac{-4}{1}=\frac{c}{a}$$
Answer: The sum is $3=-b/a$ and the product is $-4=c/a$.
Practice

Find the degree of $7u^6-\frac{3}{2}u^4+4u^2+u-8$.

Hint: Look for the largest exponent of $u$.
Key step: The largest exponent is $6$.
Solution: The degree is $6$.
Practice

Find the zeroes of $x^3-4x$.

Hint: Factor the polynomial.
Key step: $x^3-4x=x(x-2)(x+2)$.
Solution: $x(x-2)(x+2)=0$, so the zeroes are $-2,0,2$.
Practice

A quadratic graph touches the $x$-axis at exactly one point. How many distinct zeroes does it have?

Hint: A zero is an $x$-coordinate of an intersection with the axis.
Key step: The touching point has one $x$-coordinate.
Solution: It has one distinct zero; algebraically, the two zeroes coincide.

Quick recap

  • The degree is the largest exponent with a non-zero coefficient; degrees $1,2,3$ are linear, quadratic and cubic.
  • A real number $k$ is a zero of $p(x)$ exactly when $p(k)=0$.
  • Graphically, zeroes are the $x$-coordinates where $y=p(x)$ meets the $x$-axis.
  • A quadratic can have two, one (coincident), or no real zeroes; a degree-$n$ polynomial has at most $n$ zeroes.
  • For $ax^2+bx+c$, the zeroes have sum $-b/a$ and product $c/a$.
  • For $ax^3+bx^2+cx+d$, the zeroes have sum $-b/a$, pairwise-product sum $c/a$, and product $-d/a$.
Lecture 2

Relationship Between Zeroes And Coefficients Of A Polynomial

Reference video · Lecture 2Relationship Between Zeroes And Coefficients Of A Polynomial

Quadratic Polynomials

Definition

Zero of a polynomial — A number $r$ is a zero of $p(x)$ if $p(r)=0$.

Quadratic zeroes and coefficients

For $p(x)=ax^2+bx+c$, where $a
e0$, if its zeroes are $\alpha$ and $\beta$, their sum is $-b/a$ and their product is $c/a$.

Quadratic relationships
$$\alpha+\beta=-\frac{b}{a},\qquad \alpha\beta=\frac{c}{a}$$

Use these relationships to verify zeroes or construct a quadratic polynomial.

Use the quadratic relationships

  1. For a given polynomial, identify $a$, $b$, and $c$.
    $$p(x)=ax^2+bx+c$$
  2. Find the zeroes by factoring or another suitable method.
    $$p(x)=0$$
  3. Calculate their sum and product, then compare with the coefficient formulas.
    $$\alpha+\beta\quad\text{and}\quad\alpha\beta$$
  4. To construct a polynomial from a sum and product, choose $a=1$ unless another leading coefficient is required.
    $$x^2-(\alpha+\beta)x+\alpha\beta$$
Worked example

Find the zeroes of $x^2+7x+10$ and verify the relationships between the zeroes and coefficients.

  1. Factor the polynomial to find its zeroes.
    $$x^2+7x+10=(x+2)(x+5)=0$$
  2. Set each factor equal to zero.
    $$x=-2\quad\text{or}\quad x=-5$$
  3. Compare the sum of the zeroes with $-b/a$.
    $$-2+(-5)=-7=-\frac{7}{1}$$
  4. Compare their product with $c/a$.
    $$(-2)(-5)=10=\frac{10}{1}$$
Answer: The zeroes are $-2$ and $-5$; their sum is $-7=-b/a$ and their product is $10=c/a$.
⚠ Common mistake

Do not drop the negative sign in the sum formula: the sum is $-b/a$, while the product is $c/a$.

Practice

Find the zeroes of $x^2-3$ and verify the coefficient relationships.

Hint: Use the difference of squares with $3=(\sqrt{3})^2$.
Key step: $x^2-3=(x-\sqrt{3})(x+\sqrt{3})$.
Solution: The zeroes are $\sqrt{3}$ and $-\sqrt{3}$. Their sum is $0=-0/1$, and their product is $-3=-3/1$.
Practice

Find a quadratic polynomial whose zeroes have sum $-3$ and product $2$.

Hint: For a monic quadratic, use $x^2-(\text{sum})x+\text{product}$.
Key step: $x^2-(-3)x+2$.
Solution: One such polynomial is $x^2+3x+2$. Any non-zero constant multiple also has the same zeroes.

Cubic Polynomials

Definition

Cubic zeroes — The zeroes of a cubic polynomial are the values of $x$ that make the polynomial equal to zero; a cubic has at most three zeroes.

Cubic zeroes and coefficients

For $p(x)=ax^3+bx^2+cx+d$, where $a
e0$, and zeroes $\alpha$, $\beta$, and $\gamma$, the sum, pairwise-product sum, and product of the zeroes follow the coefficient relationships below.

$$\alpha+\beta+\gamma=-\frac{b}{a},\qquad \alpha\beta+\beta\gamma+\gamma\alpha=\frac{c}{a},\qquad \alpha\beta\gamma=-\frac{d}{a}$$

Verify cubic zeroes and coefficients

  1. Identify the coefficients $a$, $b$, $c$, and $d$.
    $$p(x)=ax^3+bx^2+cx+d$$
  2. Substitute each proposed zero into $p(x)$ and check that the result is zero.
    $$p(\alpha)=p(\beta)=p(\gamma)=0$$
  3. Calculate the sum, the sum of pairwise products, and the product of the zeroes.
    $$\alpha+\beta+\gamma,\quad\alpha\beta+\beta\gamma+\gamma\alpha,\quad\alpha\beta\gamma$$
  4. Compare these values with the three coefficient relationships.
    $$-\frac{b}{a},\quad\frac{c}{a},\quad-\frac{d}{a}$$
Worked example

Verify that $3$, $-1$, and $-\frac13$ are the zeroes of $3x^3-5x^2-11x-3$, then verify the coefficient relationships.

  1. Substitute the first two proposed zeroes into the polynomial.
    $$p(3)=81-45-33-3=0,\quad p(-1)=-3-5+11-3=0$$
  2. Check the third proposed zero; the cubic term is negative because $3(-1/3)^3=-1/9$.
    $$p\left(-\frac13\right)=-\frac19-\frac59+\frac{11}{3}-3=-\frac23+\frac{11}{3}-3=0$$
  3. Calculate the sum of the zeroes and compare with $-b/a$.
    $$3+(-1)-\frac13=\frac53=-\frac{-5}{3}$$
  4. Calculate the sum of pairwise products and compare with $c/a$.
    $$3(-1)+(-1)\left(-\frac13\right)+\left(-\frac13\right)3=-3+\frac13-1=-\frac{11}{3}$$
  5. Calculate the product and compare with $-d/a$.
    $$3(-1)\left(-\frac13\right)=1=-\frac{-3}{3}$$
Answer: The three substitutions give zero, and the sum, pairwise-product sum, and product are $\frac53$, $-\frac{11}{3}$, and $1$, matching $-b/a$, $c/a$, and $-d/a$.
Practice

For $2x^3-5x^2-14x+8$, verify the coefficient relationships if the zeroes are $4$, $-2$, and $\frac12$.

Hint: Calculate the zeroes' sum, pairwise-product sum, and product.
Key step: $4+(-2)+\frac12=\frac52$; the pairwise-product sum is $-7$ and the product is $-4$.
Solution: The values are $\frac52=-b/a$, $-7=c/a$, and $-4=-d/a$, so all three relationships hold.

Quick recap

  • For $ax^2+bx+c$, the sum of the zeroes is $-b/a$ and their product is $c/a$.
  • A monic quadratic with zeroes' sum $S$ and product $P$ is $x^2-Sx+P$.
  • For $ax^3+bx^2+cx+d$, the sum of the zeroes is $-b/a$.
  • The sum of pairwise products of cubic zeroes is $c/a$.
  • The product of cubic zeroes is $-d/a$.

Formula sheet

Quadratic Zeroes and Coefficients
$$\alpha+\beta=-\frac{b}{a},\qquad \alpha\beta=\frac{c}{a}$$
Cubic Zeroes and Coefficients
$$\alpha+\beta+\gamma=-\frac{b}{a},\qquad \alpha\beta+\beta\gamma+\gamma\alpha=\frac{c}{a},\qquad \alpha\beta\gamma=-\frac{d}{a}$$
Quadratic relationships
$$\alpha+\beta=-\frac{b}{a},\qquad \alpha\beta=\frac{c}{a}$$
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Social Science — ALL CHAPTERS

1

RESOURCES AND DEVELOPMENT

2

FOREST AND WILDLIFE RESOURCES

3

WATER RESOURCES

4

AGRICULTURE

5

MINERALS AND ENERGY RESOURCES

6

MANUFACTURING INDUSTRIES

7

LIFELINES OF NATIONAL ECONOMY

1

POWER SHARING

2

FEDERALISM

3

GENDER, RELIGION AND CASTE

4

POLITICAL PARTIES

5

OUTCOMES OF DEMOCRACY

1

THE RISE OF NATIONALISM IN EUROPE

2

NATIONALISM IN INDIA

3

THE MAKING OF A GLOBAL WORLD

4

THE AGE OF INDUSTRIALISATION

5

PRINT CULTURE AND THE MODERN WORLD

1

DEVELOPMENT

2

SECTORS OF THE INDIAN ECONOMY

3

MONEY AND CREDIT

4

GLOBALISATION AND THE INDIAN ECONOMY

5

CONSUMER RIGHTS