Polynomials Grade 10

[Pages:10]ID : lk-10-Polynom ials [1]

Grade 10 Polynomials

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Choose correct answer(s) from the given choices

(1) Find the sum of the zeroes of the polynomial q2 - q - 56.

a. -1

b. 2

c. 1

d. 0

(2)

If

the

product

of

the

zeros

of

the

polynomial

2

ax

-

61x

+

7

is

7

, what is the value of a ?

40

a. 43

b. 40

c. 44

d. 42

11 (3) If a and b are the zeros of quadratic polynomial x2 + 3px - q, find the value of + .

a b

-6p a.

q

3p b.

q

-3p c.

q

6p d.

q

(4)

Find

the

zeros

of

the

polynomial

f(x)

=

3

x

-

2x2

-

25x

+

50,

if

it

is

given

that

two

of

its

zeros

are

equal in magnitude but opposite in sign.

a. 2, 6 and -5

b. 5, -5 and 2

c. 5, -5 and -2

d. 2, -3 and 5

Answer the questions

(5) Find a quadratic polynomial whose zeros are reciprocals of the zeros of the polynomial

2

x

-

4x

+

3.

(6)

Find

the

zeros

of

the

polynomial

f(x)

=

3

x

-

8x2

+

17x

-

10,

if

it

is

given

that

sum

of

its

two

zeros

is 6.

(7) Find the sum of the roots of the quadratic equation y2 - y - 6 = 0

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ID : lk-10-Polynom ials [2]

(8) Verify that 2, 7, and 5 are the zeros of the cubic polynomial 6x3 - 59x2 + 129x - 70 and hence 6

verify the relation between its zeros and coefficients.

(9) If two zeros of polynomial x3 + bx2 + cx + d are 5+3 and 5-3, find its third zero.

(10) Find the sum of the zeroes of the polynomial q2 + 3q - 28.

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Solutions

(1) c. 1

ID : lk-10-Polynom ials [3]

Step 1

We know that the sum of the zeroes, or the roots of any 2nd degree polynomial of the form aq2 + bq

-b + c is equal to .

a

Step 2

Comparing the given polynomial with the standard form ax2 + bx + c. We have, b = -1, and a = 1.

-b

-(-1)

Therefore, the value of

in this case will be

, which is equal to 1.

a

1

Step 3 Hence, the sum of the zeroes of the given polynomial is equal to 1.

(2) b. 40

Step 1

Let

and

be

the

zeros

of

the

polynomial

2

ax

-

61x

+

7.

constant term 7

Then, =

coefficient of x2

=

.

a

7

Also, we are given that = .

40

77

Thus, =

a 40

Step 2

Hence, a = 40.

a = 40.

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3p (3) b.

q

Step 1

3p

Since, sum of zeros = a + b = -

= -3p

1

Step 2

-q

Since, product of zeros = ab =

= -q

1

Step 3

1 1 a + b 3p

Therefore, + =

=

.

a b

ab

q

ID : lk-10-Polynom ials [4]

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(4) b. 5, -5 and 2

Step 1

On

comparing

the

polynomial

3

x

-

2x2

-

25x

+

50,

with

the

standard

form

3

ax

+

2

bx

+

cx

+

d

=

0,

we

get:

a=1 b = -2 c = -25 d = 50

Step 2

Let , and be the zeros of polynomial.

Step 3

It is given than = -

Step 4 Sum of zeros,

-b ++ =

a

2 ++ =

1 + (-) + = 2 =2

Step 5 Product of zeros,

d

=

a

50 = 1

()(-)(2) = 50

2

=

50

-2

2

=

-25

= 5 or - 5

Step 6

Now, = -5 or 5

Step 7

Hence, zeros are 5, -5 and 2.

ID : lk-10-Polynom ials [5]

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(5)

k[x2 -

4 x+

1]

33

ID : lk-10-Polynom ials [6]

Step 1

Let

and

be

the

zeros

of

polynomial

2

x

-

4x

+

3 .

Step 2

-4 Since, sum of zeros + = -

1 =4

Step 3

3 Since, product of zeros =

1 =3

Step 4 Let S and P respectively be the sum and products of zeros of the required polynomial.

Step 5

11 S= +

+ =

4 =

3

Step 6

11 P = ( )( )

1 =

1 =

3

Step 7

Thus, required polynomial will be k[x2 - Sx + P ] = [k x2 -

4 3x+

1 3 ]

( where k is a constant ).

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(6) 5, 1 and 2

ID : lk-10-Polynom ials [7]

Step 1

Consider

the

given

equation

3

x

-

8x2

+

17x

-

10,

let

,

,

and

be

the

zeros

of

the

polynomial.

Step 2

Given that the product of its two zeroes is 6. + = 6

Step 3

-b Sum of the roots =

a

++ =8 =8-6 =2

Step 4

-d Product of zeros =

a

-(-10) =

1 = 10

10 =

10 =

2 = 5

Step 5

(6 - ) = 5

6

-

2

=

5

2

-

6

+

5

=

0

( - 5) ( - 1) = 0

= 5 or 1

Step 6

If = 5 = 6- =6-5 =1

If = 1 = 6- =6-1 =5

Step 7

Hence, zeros are 5, 1 and 2.

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(7) 1

Step 1

b For quadratic equation ay2 + by + c = 0, the sum of roots is - .

a

Step 2 For given equation y2 - y - 6 = 0, a = 1, b = -1 and c = -6

Step 3 Therefore sum of roots,

-1 ? b S =

a -1 ? -1 S =

1 S = 1

ID : lk-10-Polynom ials [8]

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