POLYNOMIALS (LECTURE 5)

POLYNOMIALS

LESSON-5
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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.

-Identify the relation between zeroes & coefficients of a cubic polynomial
-Apply the above relation to construct  a polynomial

Please write the following learning outcomes in your note books 

I will be able to:

• Apply the relationship between zeroes and coefficients of a polynomial to various forms of algebraic expressions.

Q1) If α & β are zeroes of 2x² - 5x + 3, then find the values of α² + β².

A1) Step-1 : a = ___, b = ___, c = ___ 

                              {Hint : write the coefficients of the given polynomial}

        Step-2 : α + β = ___, αβ = ___ 

                              {Hint : use the relationship equations}

        Step-3 : (α + β)² = α² + β² + 2αβ 

                      ⇒(α + β)² - 2αβ = α² + β²   → (equation 1)

                              {Hint : use the algebraic identity and then transfer 2αβ to LHS}

        Step-4 : ⇒(___)² - 2(___) = α² + β² 

                              {Hint : substitute values from Step-2 in equation 1 & solve}

Q2) If α & β are zeroes of 2x² - 5x + 3, then find the values of 1/2α + 1/2β.

A2) Step-1 : a = ___, b = ___, c = ___ 

                              {Hint : write the coefficients of the given polynomial}

        Step-2 : α + β = ___, αβ = ___ 

                              {Hint : use the relationship equations}

        Step-3 : 1/2α + 1/2β = (β + α)/2αβ → (equation 1)

                              {Hint : use the LCM}

        Step-4 : ⇒1/2α + 1/2β = (___)/2(__) 

                              {Hint : substitute values from Step-2 in equation 1 & solve}

Q3) If α & β are zeroes of 2x² - 5x + 3, then find the values of α/β + β/α.

A3) Step-1 : a = ___, b = ___, c = ___ 

                              {Hint : write the coefficients of the given polynomial}

        Step-2 : α + β = ___, αβ = ___ 

                              {Hint : use the relationship equations}

        Step-3 : α/β + β/α = (α² + β²)/αβ → (equation 1)

                              {Hint : use the LCM}

        Step-4 : ⇒α/β + β/α = (_____)/(__)  

                              {Hint : substitute value of αβ from Step-2 & value of (α² + β²) from  

                                Q1 & solve}

Q4) If α & β are zeroes of 2x² - 5x + 3, then find the values of α³ + β³.

A4) Step-1 : a = ___, b = ___, c = ___ 

                              {Hint : write the coefficients of the given polynomial}

        Step-2 : α + β = ___, αβ = ___ 

                              {Hint : use the relationship equations}

        Step-3 : (α + β)³ = α³ + β³ + 3αβ(α + β) 

                      ⇒(α + β)³ - 3αβ(α + β) = α³ + β³ → (equation 1)

                              {Hint : use the algebraic identity and then transfer 3αβ(α + β) to 

                                                                                                      LHS}

        Step-4 : ⇒(___)³ - 3(___)(___) = α³ + β³ 

                              {Hint : substitute values from Step-2 in equation 1 & solve}

Q5) If α & β are zeroes of 2x² - 5x + 3, then find the values of α⁴ + β⁴.

A5) Step-1 : a = ___, b = ___, c = ___ 

                              {Hint : write the coefficients of the given polynomial}

        Step-2 : α + β = ___, αβ = ___ 

                              {Hint : use the relationship equations}

        Step-3 : (α + β)⁴ = ((α + β)²)² 

                                  = (α² + β² + 2αβ)² 

       ⇒(α + β)⁴ = α⁴ + β⁴ + 4α²β² + 4α³β + 4αβ³ + 2α²β²

       ⇒(α + β)⁴ = α⁴ + β⁴ + 6(αβ)² + 4αβ(α² + β²)

       ⇒(α + β)⁴ - 6(αβ)² - 4αβ(α² + β²)= α⁴ + β⁴→ (equation 1)

                              {Hint : use the algebraic identity and then transfer 

6(αβ)² , 4αβ(α² + β²) to LHS}

        Step-4 : ⇒(____)⁴ - 6(__)² - 4(__)(____)= α⁴ + β⁴

                              {Hint : substitute values from Step-2 & from Q1, in equation 1 & 

                                                solve}

Q6) If α & β are zeroes of 2x² - 5x + 3, then find the values of α - β.

A6) Step-1 : a = ___, b = ___, c = ___ 

                              {Hint : write the coefficients of the given polynomial}

        Step-2 : α + β = ___, αβ = ___ 

                              {Hint : use the relationship equations}

        Step-3 : (α - β)² = α² + β² - 2αβ

                                        {Hint : use the algebraic identity}

                   

                      ⇒(α - β)² = α² + β² + 2αβ - 2αβ - 2αβ 

                                            {Hint : add 2αβ and subtract 2αβ}

                      

                      ⇒(α - β)² = (α² + β² + 2αβ) - 4αβ

                              {Hint : Rearranging}

                                 ⇒(α - β)² = (α + β)² - 4αβ  → (equation 1)

                                              {Hint : use the algebraic identity}

        Step-4 : ⇒(α - β)² = (___)² - 4(___) 

                              {Hint : substitute values from Step-2 in equation 1 & solve. After that take square root of the obtained value to get the value for (α - β)}

Tomorrow i  am going to take a google meet  with you to explain lecture 4 & 5. plz be on time. link i will give you through whatsapp. tomorrow have single period,so  till 7:35am.

HOME WORK: 

Q7)  If α & β are zeroes of x² - x - 6, then find the values of :

(i) α² + β²

(ii) 1/2α + 1/2β

(iii) α/β + β/α

(iv) α³ + β³

(v) α⁴ + β⁴

(vi)  α - β

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