<span style="color: rgb(0, 0, 0); font-family: Roboto, sans-serif; font-size: 14px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial; display: inline !important; float: none;">If the zeros of the polynomial x 3 - 3 x 2 + x + 1 <span style="color: rgb(0, 0, 0); font-family: Roboto, sans-serif; font-size: 14px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial; display: inline !important; float: none;"> are ( a - b ) , a and ( a + b ) , find the values of a and b .

Given polynomial is p ( x ) = x 3 − 3 x 2 + x + 1 Let α = ( a − b ) , β = a and γ = a + b Now, α + β + γ = − ( − 3 ) 1 ⇒ ( a − b ) + a + ( a + b ) = 3 ⇒ 3 a = 3 ⇒ a = 1 Also, α β + β γ + γ α = 1 1 ⇒ ( a − b ) a + a ( a + b ) + ( a + b ) ( a − b ) = 1 ⇒ a 2 − a b + a 2 + a b + a 2 − b 2 = 1 ⇒ 3 a 2 − b 2 = 1 ⇒ 3 ( 1 ) 2 − b 2 = 1 ⇒ b 2 = 2 ⇒ b = ± 2 Hence, a = 1 and b = ± 2 .

<span style="color: rgb(0, 0, 0); font-family: Roboto, sans-serif; font-size: 14px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial; display: inline !important; float: none;">Find the quadratic polynomial, the sum of whose zeros is - 5 and their product is 6 .

Let α and β be the zeros of the required quadratic polynomial. Then, we have α + β = - 5 and α β = 6 Now, a quadratic polynomial whose zeros are α and β is given by p ( x ) = k x 2 − ( α + β ) x + α β where k is any real number. Hence, required quadratic polynomial, p ( x ) = k x 2 − ( − 5 ) x + ( 6 ) = x 2 + 5 x + 6 . when k = 1

<span style="color: rgb(0, 0, 0); font-family: Roboto, sans-serif; font-size: 14px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial; display: inline !important; float: none;">Find a cubic polynomial whose zeros are 3 , 5 and - 2 .

Let α , β and γ be the zeros of the required cubic polynomial. Then, we have α = 3 , β = 5 and γ = - 2 Now, α + β + γ = 3 + 5 - 2 = 6 α β + β γ + γ α = 3 × 5 + 5 × - 2 + - 2 × 3 = 15 - 10 - 6 = - 1 α β γ = 3 × 5 × - 2 = - 30 Now, a cubic polynomial whose zeros are α , β and γ is given by p ( x ) = k x 3 − ( α + β + γ ) x 2 + ( α β + β γ + γ α ) x − α β γ , where k is any real number. Hence, required cubic polynomial, p ( x ) = k x 3 − ( 6 ) x 2 + ( − 1 ) x − ( − 30 ) = x 3 − 6 x 2 − x + 30 . when k = 1

<span style="color: rgb(0, 0, 0); font-family: Roboto, sans-serif; font-size: 14px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial; display: inline !important; float: none;">Using remainder theorem, find the remainder when p ( x ) = x 3 + 3 x 2 - 5 x + 4 <span style="color: rgb(0, 0, 0); font-family: Roboto, sans-serif; font-size: 14px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial; display: inline !important; float: none;"> is divided by ( x - 2 ) .

Here, p ( x ) = x 3 + 3 x 2 - 5 x + 4 By the remainder theorem, we know that when p ( x ) is divided by ( x − 2 ) , the remainder is p ( 2 ) . Now, p ( 2 ) = ( 2 ) 3 + 3 ( 2 ) 2 − 5 ( 2 ) + 4 = 8 + 12 − 10 + 4 = 14 Hence, the required remainder is 14 .

<span style="color: rgb(0, 0, 0); font-family: Roboto, sans-serif; font-size: 14px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial; display: inline !important; float: none;">Show that ( x + 2 ) is a factor of f ( x ) = x 3 + 4 x 2 + x − 6 .

By the factor theorem, ( x + 2 ) will be a factor of f ( x ) if f ( − 2 ) = 0 . Now, f ( x ) = x 3 + 4 x 2 + x − 6 ⇒ f ( − 2 ) = ( − 2 ) 3 + 4 ( − 2 ) 2 + ( − 2 ) − 6 = − 8 + 16 − 2 − 6 = 0 Hence, ( x + 2 ) is factor of f ( x ) = x 3 + 4 x 2 + x − 6 .