POLYNOMIALS (LECTURE 4)

POLYNOMIALS

LESSON-4, 

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Learning Outcomes covered so far: 

-recall the terms and definitions related to algebra.

-The geometrical representations of linear and quadratic polynomials and the geometrical meaning of their zeroes.

-find the zeroes of a quadratic polynomial 

-verify the relationship between  zeroes and the coefficients .

-recall what is sum and product of zeroes of a quadratic polynomial.

-Form a quadratic polynomial when sum and product of zeroes are given.

Please write the following learning outcomes in your note books 

I will be able to:

• identify the relation between zeroes & coefficients of a cubic polynomial
• apply the above relation to construct  a polynomial

Observe the image given below👇

Q1

Answer the following questions based on the above example:

Q2) The zeroes (from what you see in Q1), for the given polynomial are α =____, β = ____ & γ = ____  

                           {Hint: The abscissa of the points where the intersects / meets the X-axis}

Q3) Simplify 2(x + 1) (x - 2)(x - 3)  

                           {Hint: Find the product}

A3)  2x³ - 8x² + 2x  + 12  {Hint: Show steps of the product}

Q4) How many different polynomials are possible with the zeroes that you have found out in Q2 ?

Q5) If a curve passes through (-1,0), (2,0), (3,0) & (4,30), then write the cubic polynomial.

A5)  Step-1: The zeroes of the cubic polynomial are:

          α =____, β = ____ & γ = ____ 

                   {Hint: The abscissa of the points where the ordinate is 0}

        Step-2: Write the factors of the polynomial, using the     

                      zeroes

        p(x) = k  (             )(           )(           )

                   {Hint: If a is a zero, then (x - a) is a factor, if  -a is a zero, then (x + a) is a factor}

        Step-3: Substitute the 4th given point (4,30) in the polynomial constructed in Step-2 & solve "k"     

                      zeroes

        30 =  k (             )(           )(           )

                   {Hint: If a is a zero, then (x - a) is a factor, if  -a is a zero, then (x + a) is a factor}

        Step-4: Substitute the value of k in the polynomial constructed in Step-2 and simplify it

              p(x) = __  (             )(           )(           )

                      =  ?

Q6) Are the zeroes given in Q2 & Q5 same ? (Yes / No) 

Q7) How many zeroes will a linear polynomial have ? (1 / 2 /3)

Q8) How many zeroes will a quadratic polynomial have ? 

                                             (1 / 2 / 3 )

Q9) How many zeroes will a cubic polynomial have ? (1 / 2 /3)

RECALL 👇

If α, β are zeroes / roots of p(x) = a x2 + bx + c, then

              

Q10) Is there a relationship between the zeroes & its coefficients for a cubic polynomial ? (Yes / No)

Very Important Note

As much the degree, 

those many the zeroes!

 As many the zeroes, 

those many relations!

Q11)  If , p(x) = 2x³ - 8x² + 2x  + 12, answer the following:

   (i)  a = ____ , b = ____ , c = ____ , d = ____

           {Hint: the coefficients of x³, x² , x & constant respectively}

   (ii)  α =____, β = ____ & γ = ____  

           {Hint: Refer to Answer 2}

     (iii)  -b/a = ____ 

           {Hint: Substitute the values from (i)}

     (iv) α + β + γ = ____ 

           {Hint: Substitute the values from (ii)}

     (v) α + β + γ = -b/a    {Hint: Compare the answers in (iii) & (iv)}

         {This is the first relation: SUM OF THE ZEROES}

           

    (vi)  c/a = ____ 

           {Hint: Substitute the values from (i)}

     (vii) αβ + βγ + γα = ____ 

           {Hint: Substitute the values from (ii)}

     (viii) αβ + βγ + γα = c/a    {Hint: Compare the answers in (vi) & (vii)}

         {This is the second relation: SUM OF THE PRODUCT OF ZEROES, TAKEN TWO AT A TIME}

    (ix)  -d/a = ____ 

           {Hint: Substitute the values from (i)}

     (x) αβγ = ____ 

           {Hint: Substitute the values from (ii)}

     (xi) αβγ = -d/a    {Hint: Compare the answers in (ix) & (x)}

         {This is the third relation: PRODUCT OF the ZEROES}

Q12) Refer to NCERT, Ex. 2.4 (Q1, (i))

Q13) Refer to NCERT, Ex. 2.4 (Q2)

  HOME WORK: EX.2.4 - Q1 (ii) (iii)