POLYNOMIALS (LECTURE 3)

lesson 3 polynomials

LESSON 3 POLYNOMIALS 

                                                                   GOOD MORNING EVERYONE!!

Few Instructions 

Text in red has to be noted down as its your class work.
 Text in blue  are  videos  (watch )

 Text in green is your homework

AQAQ

  The polynomial of type ax2 + bx + c, a = 0 is of type

      (a)   linear

(b)   quadratic

(c)   cubic

(d)   Biquadratic

In the previous class we discussed about 

1. CLASSIFY THE POLYNOMIALS ON THE BASIS OF NUMBER OF TERMS AND DEGREE.
2. COMPREHEND AND MEMORIZE ABOUT SOME SPECIAL POLYNOMIALS.

TODAY'S LEARNING OUTCOMES:
I WILL BE ABLE TO:

1. EVALUATE  VALUE OF A  POLYNOMIAL
2.EVALUATE  ZERO  OF A  POLYNOMIAL.

Value  of a Polynomial

For Example: 
Find  value  of this polynomialp(x) = x + 2  at x =1

.(step i)     Consider p(x) = x +1 .

,( step ii)   If we put x = 1  in p(x), we get
  
 (step (iii)   p(1) = 1 + 2 = 3


Thus,3  is  the value  of the polynomial p(x).   at  x = 1


 Zero of a Polynomial : The value of variable for which the polynomial becomes zero is called as the zero of the polynomial
.
For Example:
Find zero of this polynomial  p(x) = x + 2. 

(stepi)   p(x) = x + 2. 

(stepii) If we put x = -2 in p(x), , we get

.(stepiii)  p(-2) = -2 + 2 = 0

Thus,( -2) is a zero of the polynomial p(x)................................  why?.

 (NOTE :- The value of polynomial p(x) has become zero , when x=(_-2 ).

TO KNOW  MORE ,CLICK THE LINK GIVEN BELOW 

ZERO OF A POLYNOMIAL

Lets  solve another  Example: (Read carefully and try to understand )

 Find value of polynomial 3a2 + 5a + 1 at a = 3

(i) Here, p(a) = 3a2 + 5a + 2.
(ii) Now, substituting a = 3, we get
,(iii) p(3) = 3 x (3)2 + 5 x 3 + 2 = 27 + 15 + 2 = 44

. Thus the value of polynomial p(x)   is 44,  if  a = 3

 Find  zero  of polynomial 3a2 + 5a + 1 .

(i) Here, p(a) = 3a2 + 5a + 2
.(ii) Now, substituting a = ( -1), we get,
(iii) p(-1 ) = 3 x (-1 )2 + 5 x (-1) + 2 = 3 - 5 + 2 = 0

. Thus the zero  of polynomial p(x)   is   (-1)

so Answer these questions 

  The value  of the polynomial p(x) = 2x + 5 is,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,(  5   or  7  ) ,  at  x = 1,,

 The  value  of p (x) =  x  + 2   is   2 ,  if  the value of x is   ..............................(  one / zero)

 The value of  p (x) =  x2 + 4x + 2  at  x = ( -1) is .................   ( -1   or  7 )

 see the video given for solving Q1   Ex 2.2  (cw)

                                      Q1EX 2.2

          solution:- 1 (iii)  p(x)=  5x-4x2 + 3  at   x=2 

 see the video given for solving Q3   Ex 2.2

Q3 EX. 2

  Now lets solve  Q3

 

                                                      CLICK THE LINK GIVEN BELOW 

                     FINDING ZERO OF A  LINEAR  POLYNOMIAL

                                 Now answer the following 

                 The zero of the polynomial p(x) = 2x + 5 is,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,(  5/2   or  -5./2  ),,

                  The number of zeros of x2 + 4x + 2  is...............................(  one   or  two) 

                    Q4  (i)

                    Let's  find  the zero of given polynomial  
                   Q4  (vi)p(x) = ax , a ≠ 0

                     since , p(x) = 0  
                   ⇒ ax = 0  
                  ⇒   x = 0    ,Thus  zero of p(x) is 0

                    (vii) p(x) = cx +d

                  since , p(x) = 0 
                   ⇒   cx +d = 0
                   ⇒ cx = -d
                    ⇒ x = -d/c, Thus zero of p(x) is  -d / c 

 NOTE -:

(i) A non-zero constant polynomial has no zero
.(ii) A linear polynomial has one and only one zero
.(iii) A zero of a polynomial might not be 0 or 0 might be a zero of a polynomial
.(iv) A polynomial can have more than one zero. (quadratic , cubic.......)

  Q1. Check whether at x = -1/7 is zero of the polynomial p(x) = 7x + 1. 

   Q2.Find zero of the polynomial p(x) = 2x+ 2. 

      Ex.2.2   Q1  (i, ii )  ,   Q2  (ii,iii,iv, ), Q3 ( ii,iii,iv,v,vi,viii) Q4  (ii,iii,iv,v )

17 SDG (Sustainable  Development Goals )

     TAKE CARE OF YOURSELF .
     HAVE A GOOD DAY.

POLYNOMIALS (LECTURE 2)

LESSON 2 POLYNOMIALS

GOOD MORNING EVERYONE!!

In the previous class we discussed about 
1What is a polynomial?
2. The terms and degree of a polynomial

TODAY'S LEARNING OUTCOMES:
I WILL BE ABLE TO:
1. CLASSIFY THE POLYNOMIALS ON THE BASIS OF NUMBER OF TERMS AND DEGREE.
2. COMPREHEND AND MEMORIZE ABOUT SOME SPECIAL POLYNOMIALS.

                               READ,VIEW AND UNDERSTAND



HOW TO CLASSIFY POLYNOMIALS ON THE BASIS OF NUMBER OF TERMS?

YOU HAVE DONE THIS IN CLASS 8 IN THE CHAPTER 
"ALGEBRAIC EXPRESSIONS"




How many terms are there in each of these? 
1.p(x) = 2x² + 5x – 3 


NO. OF TERMS= 3



2.q(x) = x¹ –1

NO. OF TERMS = 2


3.r(y) = y³  

NO. OF TERMS = 1

4.s(u) = 2 – u – u² + 6u⁵

NO. OF TERMS = 4

In the above examples, p(x) is known as trinomial
                                       q(x) is known as binomial
                                       r(y) is known as monomial
                                       s(u) is known as quadrinomial


                                     

HOW TO CLASSIFY POLYNOMIALS ON THE BASIS OF DEGREE?

You learnt about the DEGREE of a polynomial in the previous class

Today we will do the classification.

watch this☟

.

⭕Now you can identify a linear ,quadratic and a cubic polynomial

look at the IMAGE below and answer the following

1.WRITE THE LINEAR POLYNOMIAL(S).

3x,2x -1

HOW MANY MAXIMUM TERMS ARE THERE IN LINEAR POLYNOMIAL?

2

2WRITE THE QUADRATIC POLYNOMIAL(S)

x² , 2x² + 5 , 3x²  + 5x +1

HOW MANY MAXIMUM TERMS ARE THERE IN QUADRATIC POLYNOMIAL?

3

3WRITE THE CUBIC  POLYNOMIAL(S)

.x³  ,  x³  +  1  ,  x³  +  1 +x  , x³  +  1  +x² +x

.

HOW MANY MAXIMUM TERMS ARE THERE IN CUBIC POLYNOMIAL?

4  

DID YOU OBSERVE?

 IN ABOVE EXAMPLES

THAT MAXIMUM NO. OF TERMS = DEGREE OF THE POLYNOMIAL +1

NOW, YOU WILL LEARN ABOUT 2 SPECIAL POLYNOMIALS

WATCH THIS ☟

                          CLASS WORK  (WRITE DOWN)

I       CLASSIFICATION OF POLYNOMIALS 

A. ON THE BASIS OF NO. OF TERMS

❄❄

A polynomial can have any (finite) number of terms.

B. ON THE BASIS OF DEGREE

MAXIMUM NUMBER OF TERMS IN A POLYNOMIAL  = DEGREE +1

II CONSTANT POLYNOMIAL

EVERY NON ZERO  CONSTANT LIKE 2 , 1/2 OR  0.5 CAN  BE WRITTEN AS  

 2xº ,1/2(xº)  or 0.5(xº)      (❄  as   xº =1)

These are known as CONSTANT polynomials . The degree of non zero  constant polynomial is 0.

III  ZERO POLYNOMIAL

The constant  0 is known as zero polynomial.

Since 0 can be written as  0  xº  / 0 x¹ /0 x² or 0x⁹

so we can say that degree of zero polynomial. is NOT DEFINED

EX 2.1

3. Give one example each of a binomial of degree 35, and of a monomial of degree 100.

Ans. The binomial of degree 35 can be x³⁵ + 9.

The monomial of degree 100 can be  t¹ºº.

5. Classify the following as linear, quadratic and cubic polynomials:

Ans. (i)x² +x

We can observe that the degree of the polynomial is 2.

Therefore, we can conclude that the polynomial is a  quadratic polynomial.

(ii) x - x³

We can observe that the degree of the polynomial  is 3.

Therefore, we can conclude that the polynomial is a cubic polynomial.

(iii) y + y ² +4

We can observe that the degree of the polynomial  is 2.

Therefore, the polynomial  is a quadratic polynomial.

                                          

HOME WORK

EX2.1

Q5 (IV) TO (VII)

WE WILL END OUR CLASS HERE!

SEE YOU FOR THE NEXT CLASS!