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.