Factor Theorem and Remainder Theorem exam questions

Questions

• 1. Show (x+2) is a factor of x^3+9x^2+26x+24
• 2. Factorise f(x)=x^3+3x^2-4x-12
• 3. What is the remainder when 2x^3-3x^2-39x+20 is divided by (x-1)
• 4. x^3+px^2-10x-19 is divided by (x+2) to give a remainder of 5. Find p
• 5. 2x^4+3x^3+5x^2+cx+d is divisible by (x-2)(x+3). Find C and D

1. Show (x+2) is a factor of x^3+9x^2+26x+24

• f(x) = x^3+9x^2+26x+24
• f(-2) = (-2)^3+9(-2)^2+26(-2)+24 , f(-2) = -8+36-52+24 = 0
• f(-2) = 0 , so by the factor theorem, (x+2) is a factor of f(x).

2. Factorise f(x)=x^3+3x^2-4x-12

• f(x)=x^3+3x^2-4x-12
• f(2) = (2)^3+2(2)^2-4(2)-12 , f(2)=-8+12+8-12 , so f(-2)=0 , and therefore (x-2) is a root
• Now use the box method to work out the quadratic remaining is (x^2+4x+4).
• Now factorise this quadratic to (x+3)(x+2) , leaving the answer (x-2)(x+2)(x+3)

3....

• What is the remainder when 2x^3-3x^2-39x+20 is divided by (x-1)
• f(1) = 2(1)^3-3(1)^2-39(1)+20 , 2-3-39+20 = -20
• Remainder is -20

4.....

• 4. x^3+px^2-10x-19 is divided by (x+2) to give a remainder of 5. Find p
• f(-2)=5 : -8+4p+20-19 = -7+4p=5 , 12=4p , 3=p

5.....

• 5. 2x^4+3x^3+5x^2+cx+d is divisible by (x-2)(x+3). Find C and D
• f(2) = 32+24+20+2c+d = 76+2c+d : 2c+d=-76
• f(-3) = 162-81+45-3c+d = 126-3c+d : 3c-d=126
• 3c-d=126 + 2c+d=-76 : 5c=50 , so c=10
• 3(10)-d=126 , 30-d=126 , 30-126=d , d=-96
• c=10 , d=-96