POLYNOMIAL EXAM QUESTIONS, Exercises of Algebra

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POLYNOMIAL

EXAM

QUESTIONS

Question 1 ()** Multiply out and simplify

( 2 x^^2 −^ x^ −^3 )^ ( 1 +^2 x^ −^ x^2 ),

writing the answer in ascending powers of x.

− 3 − 7 x + 3 x^2^ + 5 x^3 − 2 x^4

Question 2 ()**

f ( x ) ≡ x^3^ − 3 x^2 + 6 x − 40.

a) Show that ( x − 5 )is not a factor of f ( x ).

b) Find a linear factor of f ( x ).

MP1-N , ( x −4)

Question 5 ()**

a) Use the factor theorem to show that ( x + 3 )is a factor of x^3 + 5 x^2 − 2 x − 24.

b) Factorize x^3 + 5 x^2 − 2 x − 24 fully.

( x^ +^3 )(^ x^ −^2 )(^ x +^4 )

Question 6 (+)** Find the coefficient of x^3 in the expansion of

( 2 x^3^^ −^5 x^2^^ +^2 x^ −^1 )^ ( 3 x^3^^ +^2 x^^2 −^9 x +^7 ).

... + 60 x^3 ...

Question 7 (+)** Multiply out and simplify

(^1 +^ x^ )(^1 +^ x^^2 )(^1 −^ x^ +^ x^2 ),

writing the answer in ascending powers of x.

1 + x^2 + x^3^ + x^5

Question 8 (+)**

a) Use the factor theorem to show that ( x − 5 )is a factor of x^3 − 19 x − 30.

b) Factorize x^3 − 19 x − 30 into three linear factors.

( x^ +^3 )(^ x^ +^2 )( x −^5 )

Question 11 (+)**

f ( x ) ≡ x^3 − 2 x^2 + kx + 6 ,

where k is a constant.

a) Given that ( x − 3 )is a factor of f ( x ), show that k = − 5.

b) Factorize f ( x )into three linear factors.

c) Find the remainder when f ( x )is divided by ( x + 3 ).

C2F , ( x − 1 )( x + 2 )( x − 3 ), R = − 24

Question 12 (+)**

a) Use the factor theorem to show that ( x + 2 )is a factor of 2 x^3 + 3 x^2 − 5 x − 6.

b) Factorize 2 x^3 + 3 x^2 − 5 x − 6 into three linear factors.

( x^ +^1 )(^ x^ +^2 )( 2 x −^3 )

Question 13 (+)**

f ( x ) ≡ 2 x^3^ − 7 x^2 − 5 x + 4

a) Find the remainder when f ( x )is divided by ( x + 2 ).

b) Use the factor theorem to show that ( x − 4 )is a factor of f ( x ).

c) Factorize f ( x )completely.

C2I , R = − 30 , ( 2 x − 1 )( x + 1 )( x − 4 )

Question 15 (+)**

f ( x ) ≡ px^3^ − 32 x^2 − 10 x + q ,

where p and q are constants.

When f ( x )is divided by ( x − 2 )the remainder is exactly the same as when f ( x )is

divided by ( 2 x + 3 ).

Show clearly that p = 8. C2J , proof

Question 16 ()* Solve the equation

x^3^ + x^2 − ( x − 1 )( x − 2 )( x − 3 ) = 12.

x = −^37 , 2

Question 17 ()*

f ( x ) ≡ 3 x^3 − 2 x^2 − 12 x + 8.

a) Find the remainder when f ( x )is divided by ( x − 4 ).

b) Given that ( x − 2 )is a factor of f ( x )solve the equation f ( x ) = 0.

C2C , R = 120 , x = −2, 23 , 2

Question 19 ()*

f ( x ) ≡ 6 x^2 + x + 7 , x ∈
.

The remainder when f ( x )is divided by ( x − a ) is the same as that when f ( x ) is

divided by ( x + 2 a ), where a is a non zero constant.

Find the value of a.

C2N , a =^16

Question 20 ()* A cubic function is defined in terms of the positive constant k as

f ( x ) ≡ x^3 + ( k − 1 ) x^2^ − k^3 , x ∈
.

It is further given that when f ( x )is divided by ( x − 3 )the remainder is 18.

a) Determine the value of k.

b) Find the remainder when f ( x )is divided by ( 2 x − 5 ).

k = 3 , (^98)

Question 21 ()* A cubic graph is defined as

f ( x ) ≡ x^3^ + x^2 − 10 x + 8 , x ∈
.

a) By considering the integer factors of 8 , or otherwise, express f ( x ) as the

product of three linear factors.

b) Sketch the graph of f ( x ).

The sketch must include the coordinates of any points where the graph of f ( x )

meets the coordinate axes.

MP1-K , f ( x ) = ( x − 2 )( x − 1 )( x + 4 )

Question 23 ()*

f ( x ) ≡ x^3^ + px^2 + qx + 6

a) Find the value of each of the constants p and q , given that …

… ( x − 1 )is a factor of f ( x )

… when f ( x )is divided by ( x + 1 ) the remainder is 8.

b) Hence solve the equation f ( x ) = 0.

C2H , p = − 2 , q = − 5 , x = 1, −2, 3

Question 24 ()*

f ( x ) ≡ 2 x^3^ − 7 x^2 − 2 x + 1

a) Use the factor theorem to show that ( 2 x + 1 )is a factor of f ( x ).

b) Find the exact solutions of the equation f ( x ) = 0.

x = − 12 , 2 ± 3

Question 25 () a)* Find the value of each of the constants a , b and c so that

6 x^3 − 7 x^2 − x + 2 ≡ ( x − 1 )( ax^2 + bx + c ).

b) Hence solve the equation 6 x^3 − 7 x^2 − x + 2 = 0. C2M , a = 6, b = −1, c = − 2 , x = −^1 2 3^ , 2 ,

Question 27 ()*

f ( x ) ≡ x^3 − 9 x^2 + 22 x − 12.

a) Show that x = 3 is a solution of the equation of the equation f ( x ) = 0.

b) Find, in exact surd form, the other two solutions of the equation f ( x ) = 0.

x = 3 ± 5

Question 28 ()*

f ( x ) ≡ x^2 − 4 x + 12.

The remainder when f ( x )is divided by ( x + k )is three times as large as when f ( x )is

divided by ( x − k ).

Determine the possible values of k. C2P , k =6, 2

Question 29 ()*

f ( x ) ≡ 2 x^3^ + kx^2 − x − 6 ,

where k is a constant

Given that f ( 3 ) = 0 , …

a) … show that k = − 5

b) … factorize f ( x )as a product of one linear and one quadratic factor.

c) … show further that, apart from x = 3 , the equation f ( x ) = 0 has no other real

f solutions. f (^) ( x (^) ) = (^) ( x − (^3) ) (^) ( 2 x^2 + x + (^2) )