Sec 3 Integrated Mathematics (IP/mainstream)
Name : _______________________________ Class (
)
Date : ___________
Remainder & Factor Theorems
Assignment
1
f ( x) and g( x) are polynomials. Jane claims that if the sum of the coefficients of f ( x) equals the sum
of the coefficients of g( x) , then the graph of y  f ( x) must intersect the graph of y  g( x) at least once.
Do you agree?
2
Given that


g( x)   x  2  3x2  4  5


h( x)   6 x  1 3x3  2 x2  x  8
find the remainder when
3
g( x)  h( x) is divided by x 1 .
(a) Write down the general form of a cubic expression.
(b) Write down the general form of a cubic equation.
4
(a) A cubic curve crosses the y-axis at 12 and the x-axis at  1, 1 and 3. Find the equation of the curve.
(b) A cubic curve crosses the y-axis at 60 and the x-axis at  2 and 5. Find the equation of the curve.
(c) A cubic curve crosses the y-axis at 4 and the x-axis at 2. Find the equation of the curve.
(a)
(b)
(c)
5
The diagram shows the curve y  x 4  2 x3  px 2  qx  r . It cuts the x-axis at  2 and 4 and the y-axis
at  8. Find the values of p, q and r and express the equation of the curve in the factorized form.
6
A polynomial f ( x) when divided by  x  2  leaves a remainder of 2x 10 . Another polynomial
2
g ( x) when divided by 3x2  5x  2 leaves a remainder of x  4 .
(i) Find a linear factor of the sum of the polynomials f ( x)  g ( x) .
(ii) Find the remainder when the product of the polynomials f ( x)  g ( x) is divided by this
linear factor.
7
The diagram shows part of the curve y  x3  ax2  bx  c . It crosses the axes at P and Q and has a
minimum point at R. If the coordinates of P, Q and R are  0, 2  ,  2, 0  and 1, 3 , evaluate a, b and c.
y
Q
P
x
R
8
A polynomial is divisible by factor x  2 but leaves a remainder of 12 when divided by x 10 .
Find the remainder when the polynomial is divided by  x  2 x  10  .
9
A cubic curve y  f ( x) intercepts the x-axis at  1, 2  5 and 2  5 .
(i) Write down the roots of f ( x)  0 .
(ii) If the coefficient of x3 is 2, express f ( x) as a cubic polynomial in x with integer coefficients.