<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;">Use remainder theorem to find the value of k , it being given that when x 3 + 2 x 2 + k x + 3 <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 - 3 ) , then the remainder is 21 .

Let p ( x ) = x 3 + 2 x 2 + k x + 3 By the remainder theorem, we know that when p ( x ) is divided by ( x − 3 ) the remainder is p ( 3 ) . Now, p ( 3 ) = ( 3 ) 3 + 2 ( 3 ) 2 + k ( 3 ) + 3 = 27 + 18 + 3 k + 3 = 48 + 3 k But, remainder = 21 ⇒ 48 + 3 k = 21 ⇒ 3 k = − 27 ⇒ k = − 9

<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;">Use remainder theorem to find the value of k , it being given that when x 3 + 2 x 2 + k x + 3 <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 - 3 ) , then the remainder is 21 .

Let p ( x ) = x 3 + 2 x 2 + k x + 3 By the remainder theorem, we know that when p ( x ) is divided by ( x − 3 ) the remainder is p ( 3 ) . Now, p ( 3 ) = ( 3 ) 3 + 2 ( 3 ) 2 + k ( 3 ) + 3 = 27 + 18 + 3 k + 3 = 48 + 3 k But, remainder = 21 ⇒ 48 + 3 k = 21 ⇒ 3 k = − 27 ⇒ k = − 9

Find the zeroes of the polynomial f ( x ) = 2 3 x 2 - 5 x + 3 and verify the relation between its zeroes and coefficients.

Given, f ( x ) = 2 3 x 2 - 5 x + 3 ⇒ 2 3 x 2 - 2 x - 3 x + 3 ⇒ 2 x ( 3 x - 1 ) - 3 3 x - 1 ⇒ 2 x - 3 or 3 x - 1 To find the zeroes of the polynomials f ( x ) = 0 , ⇒ ( 3 x - 1 ) = 0 o r ( 2 x - 3 ) = 0 ⇒ x = 1 3 o r x = 3 2 Hence the zeroes of the polynomial are 1 3 and 3 2 . Sum of zeroes = 1 3 + 3 2 = 2 + 3 2 3 = 5 2 3 = - ( ( coefficient of x ) ) ( ( coefficient ofx 2 ) ) Product of zeroes = 1 3 × 3 2 = 1 2 = ( constant term ) ( ( coefficient of x 2 ) ) Hence, the relation between its zeroes and coefficients is verified.

E M B I B E

Secondary School Mathematics for Class 10>Chapter 2 - Polynomials>TEST YOURSELF>Q 20

MEDIUM

10th CBSE

IMPORTANT

Earn 100

Use remainder theorem to find the value of $k$ , it being given that when $x^{3} + 2 x^{2} + k x + 3$ is divided by $(x - 3)$ , then the remainder is $21$ .

Important Questions on Polynomials

MEDIUM

10th CBSE

IMPORTANT

Secondary School Mathematics for Class 10>Chapter 2 - Polynomials>Exercise 2A>Q 1

Find the zeros of the polynomial

f (x) = x^{2} + 7 x + 12

and verify the relation between its zeros and coefficients.

MEDIUM

10th CBSE

IMPORTANT

Secondary School Mathematics for Class 10>Chapter 2 - Polynomials>Exercise 2A>Q 2

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients

x^{2} - 2 x - 8

MEDIUM

10th CBSE

IMPORTANT

Secondary School Mathematics for Class 10>Chapter 2 - Polynomials>Exercise 2A>Q 3

Find the zeroes of the quadratic polynomial

f (x) = x^{2} + 3 x - 10

and verify the relation between its zeroes and coefficients.

MEDIUM

10th CBSE

IMPORTANT

Secondary School Mathematics for Class 10>Chapter 2 - Polynomials>Exercise 2A>Q 4

Find the zeros of the quadratic polynomial

f (x) = 4 x^{2} - 4 x - 3

and verify the relation between its zeros and coefficients.

MEDIUM

10th CBSE

IMPORTANT

Secondary School Mathematics for Class 10>Chapter 2 - Polynomials>Exercise 2A>Q 5

Find the zeroes of the quadratic polynomial $f (x) = 5 x^{2} - 4 - 8 x$ and verify the relationship between the zeroes and coefficients of the given polynomial.