FACTOR THEOREM PRACTICE QUESTIONS

Problem 1 :

Determine whether (x+1) is a factor of the following polynomials.

(i)  6x⁴+7x³-5x-4

(ii)  2x⁴+9x³+2x²+10x+15

(iii)  3x³+8x²-6x-5

(iv)  x³-14x²+3x+12

Problem 2 :

Determine whether (x+4) is a factor of

x³ + 3x² - 5x + 36

Problem 3 :

Using factor theorem show that (x-1) is a factor of  

4x³-6x²+9x-7

Problem 4 :

Determine whether (2x+1) is a factor of

4x³+4x²-x-1

Problem 5 :

Determine the value of p if (x+3) is a factor of

x³-3x²-px+24

Problem 6 :

Find the value of a if the division of ax³ + 9x² + 4x - 10 by x + 3 leaves the remainder of 5.

Problem 7 :

If -1 and 1 are two real roots of the polynomial function

f(x) = ax³ + bx² + cx + d

and (0, 3) is the -y intercept of graph of f , what is the value of b?

A) -3      B) -1      C) 2      D) 4

Problem 8 :

If x + 3 is a factor of f(x) = ax² + bx - 15 in which a and b are constants, what is the value of 3a - b ?

Problem 8 :

f(x) = -2(x² + 7x - 3) - a(x + 2) + 1

In the polynomial f(x) is defined above, a is constant. If f(x) is divisible by x, what is the value of a ?

A)  -5/2   B)  -3   C)  7/2    D)  5

(i)  Solution :

6x⁴+7x³-5x-4

By factor theorem, if p(-1)  =  0, then (x+1) is a factor of

p(x) =  6x⁴+7x³-5x-4

p(-1)  = 6(-1)⁴+7(-1)³-5(-1)-4

=  6-7+5-4

p(-1)  =  0

(x+1) is a factor of the given polynomial.

(ii)  Solution :

2x⁴+9x³+2x²+10x+15

By factor theorem, if p(-1) = 0, then (x+1) is a factor of 

p(x)  =  2x⁴+9x³+2x²+10x+15

p(-1)  =  2(-1)⁴+9(-1)³+2(-1)²+10(-1)+15

=  2-9+2-10+15

=  0

So, (x+1) is a factor of the given polynomial.

(iii)  Solution :

3x³+8x²-6x-5

By factor theorem, if p(-1) = 0, then (x+1) is a factor of 

p(x)  =  3x³+8x²-6x-5

p(-1)  =  3(-1)³+8(-1)²-6(-1)-5

=  -3+8+6-5

p(-1)   ≠  0

(x+1) ix not the factor of 3x³+8x²-6x-5.

(iv)  Solution :

x³-14x²+3x+12

By factor theorem, if p(-1) = 0, then (x+1) is a factor of

p(x)  =  x³-14x²+3x+12

p(-1)  =  (-1)³-14(-1)²+3(-1)+12

p(-1)  =  -1-14-3+12

p(-1)  ≠  0

(x+1) is not the factor of x³-14x²+3x+12.

(2)  Solution :

By factor theorem, if p(-4) = 0, then (x+4) is a factor of

p(x)  =  x³ + 3x² - 5x + 36

p(-4)  =  (-4)³+3(-4)²-5(-4)+36

=  -64+48+20+36

p(-4)  ≠  0

(x+4) is not the factor of x³+3x²-5x+36.

(3)  Solution :

By factor theorem, if p(1) = 0, then (x-1) is a factor of 

p(x)  =  4x³-6x²+9x-7

p(1)  =  4(1)³-6(1)²+9(1)-7

=  4-6+9-7

p(1)  =  0

(x-1) is the factor of 4x³-6x²+9x-7.

(4)  Solution :

By factor theorem, if p(-1/2) = 0, then (2x+1) is a

Factor of p(x)  =  4x³+4x²-x-1

p(-1/2)  =  4(-1/2)³+4(-1/2)²-(-1/2)-1

=  -1/2+1+1/2-1

p(-1/2)  =  0

(2x+1) is the factor of 4x³+4x²-x-1.

(5)  Solution :

By factor theorem, if p(-3) = 0, then (x+3) is a 

factor of p(x)  =  x³-3x²-px+24

p(-3)  =  (-3)³-3(-3)²- p(-3)+24

This implies that  -27-27+3p+24  =  0

-30 + 3p  =  0

3p  =  30

p  =  10

So, the value of p is 10.

(6) Solution :

Given that p(x) = ax³ + 9x² + 4x - 10

Dividing p(x) by x + 3

x + 3 = 0

x = -3

p(-3) = 5

Applying the value of x in the given polynomial, we get

p(-3) = a(-3)³ + 9(-3)² + 4(-3) - 10

5 = -27a + 81 - 12 - 10

5 = -27a + 81 - 22

5 = -27a + 59

5 - 59 = -27a

27a = -54

a = -54/27

a = -2

So, the value of a is -2.

(7) Solution :

Since -1 and 1 are solutions of the polynomial,

f(x) = ax³ + bx² + cx + d

we apply these values one by one f(-1) = 0 and f(1) = 0

f(-1) = a(-1)³ + b(-1)² + c(-1) + d

0 = -a + b - c + d

-a + b - c + d = 0 -----(1)

f(1) = a(1)³ + b(1)² + c(1) + d

0 = a + b + c + d

a + b + c + d = 0 -----(2)

(1) + (2)

b + d = 0

Applying the point (0, 3) in the equation given

3 = d

Applying d = 3, we get 

b = -3

So, the answer is option A.

(8)  Solution :

Since x + 3 is a factor of the given polynomial, then x = -3 is the solution.

Applying -3 in the place of x, we get

f(-3) = a(-3)² + b(-3) - 15

0 = 9a - 3b - 15

9a - 3b = 15

Dividing by 3, we get 

3a - b = 5

So, the value of 3a - b is 5.

(9) Solution :

f(x) = -2(x² + 7x - 3) - a(x + 2) + 1

= -2x² - 14x + 6 - ax - 2a + 1

= -2x² - 14x - ax - 2a + 7

= -2x² - x(14 + a) - 2a + 7

Since the polynomial is divisible by x,

-2a + 7 = 0

2a = 7

a = 7/2

So, answer is option C.

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