CBSE

Polynomial - Concept

A polynomial is a mathematical expression consisting of the sum of terms, each term including a variable or variables raised to a power and multiplied by a coefficient.

 Zeroes of a Polynomial.

• Mathematically: K is said to be zero of a polynomial p(x) if p(k) = 0
• Graphically: Values of x coordinates where Polynomial P(x) crosses/touches the x-axis are called zeros of polynomial P(x).

Some Graphs:

•     Graph of a linear polynomial ax + b is a straight line.
•     Graph of a quadratic polynomial p(x) = ax2+ bx + c is a parabola open upwards like ∪, if a > 0.
•     Graph of a quadratic polynomial p(x) = ax2+ bx + c is a parabola open downwards like ∩, if a > 0.
•     In general, a polynomial p(x) of degree n crosses the x-axis at at-most n points.

The relationship between zeroes and the coefficients of a Quadratic Polynomial :

•     If α, β are zeroes P(x) = ax2 + bx + c, then

•     If α, β are roots of a quadratic polynomial p(x), then.

P(x) =K {x2 – (sum of roots) x + product of roots}

P(x) = K {x2 + (α+ β) x +(αβ)}

 
The relationship between the zeroes and the coefficients of a Cubic Polynomial

If α, β and γ are zeroes of a cubic polynomial p(x),

P(x) = K {x3 – (sum of zeroes) x2 + (sum of product of zeroes) x – (product of zeroes)}

P(x) = K {x3 – (α + β + γ) x2 + (αβ + βγ + αγ) x – (αβγ)}

Electric Potential:

• An object rose against gravity, gains some gravitational potential energy proportional to the work done required to raise the object against the gravitational pull. Similarly, in an electric field, when a charged object is moved against the electric field it gains some amount of energy which is defined as the electric potential energy.
• The magnitude of this electric potential energy depends on the amount of work done on the object in moving the object from one point to another against the electric field.
• NOTE: - It is assumed that electric potential of a charge at infinity is zero.
• Definition: An electric potential (also called the electric field potential) is the amount of work needed to move a unit positive charge from infinity to a specific point inside the electric field.
• If work done in moving a positive charge Q from infinity to a point in an electric field is W, then   Electric Potential (V)=Work Done(W)/ Charge (Q)
• Units of Electric Potetial= Joule/coulomb or volt (v)

BODMAS 

Bracket followed by oder of division, multiplication, addition and subtraction 

Eg;  

If 30-(15-12)+(18/2)

1)  (15-12) according to BODMAS rule 

First solve the sum inside the bracket and calculate it one orders manner

(15-12)= 3

(18/2) = 9

Thn sub in 30-(15-12)+(18/2)

=30-(3)+(9)

=27+9 (or)  30+6

=36

According to BODMAS rule answer is 36

Try it with n number if examples 

1)( 42-18)-3(18-2)+16-18/6

Try

Activity: When we dissolve salt/sugar in water, the particles of salt/sugar get into the spaces between particles of water.

Equipment’s Required: - Beaker, Water, Salt/ Sugar, and a glass rod.

Working:

• Take a 100 mL beaker.
• Fill half the beaker with water and mark the level of water.
• Dissolve some salt/ sugar with the help of a glass rod.
• Notice the water level.

Observations:

• When sugar/salt is dissolved in water, the particles of sugar disappear in water. This happens because particles of sugar/salt get adjusted in the spaces between the particles of water.
• Additionally, you will notice that there is no rise of water level takes place when one or two teaspoons of sugar is added in a glass of water, this is because sugar particles get adjusted in the space between the particles of water and no rise in the water level comes in result.
• This activity shows us that Matters are made of small particles. And there is space between these particles.

The particles of matter are very small –they are small beyond our imagination!!!!

Let us observe an activity,

Activity: Take 2-3 crystals of potassium permanganate and dissolve them in 100 mL of water.

Equipment’s Required: - Beaker, 100ml-Water, crystals of potassium permanganate, & a glass rod.

Working: -

• Take 2-3 crystals of potassium permanganate and dissolve them in 100 mL of water. Note- We can also use Dettol instead of Potassium Permanganate.
• Take out approximately 10 mL of this solution and put it into 90 mL of clear water.
• Take out 10 mL of this solution and put it into another 90 mL of clear water.
• Keep diluting the solution like this, 5 to 8 times.

Conclusion:

• we conclude that there must be millions of tiny particles in just one crystal of potassium permanganate, which keeps on dividing themselves into smaller and smaller particles. Ultimately a stage is reached when the particles cannot divide further into smaller particles.

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Are Particles of Matter Attract Each Other?

YES, Particles of matter attract each other.

Let us observe some activities,

Activity: Make four groups and form a human chain

Group1: The First group should hold each other from the back and lock arms like Idu-Mishmi dancers.

Group2: The Second group should hold hands to form a human chain.

Group3: The Third group should form a chain by touching each other with only their fingertips.

Group4: The fourth group of students should not hold hands or bounded even with their fingertips.

Observations:

• The fourth group of students was the easiest to break. This is because students are not holding hands or bounded even with their fingertips.

Conclusions:

• There was no bonding force present in the fourth group.

The French philosopher Auguste Comte (1798–1857)—often called the “father of sociology”—first used the term “sociology” in 1838 to refer to the scientific study of society. He believed that all societies develop and progress through the following stages: religious, metaphysical, and scientific.Auguste Comte is called the father of sociology because he coined the word 'Sociology' in 1830, for that branch of science which studied human behaviour. In fact, he created a hierarchy of sciences in which he put sociology at the top. He argued that sciences dealing with simple phenomena were first to arrive.