EQUATING COEFIFICIENTS OF POLYNOMIALS WORKSHEET

Problem 1 :

Find constants a, b and c given that

2x² + 4x + 5 = ax² + [2b - 6]x + c for all x

Solution

Problem 2 :

2x³ - x² + 6 = (x - 1)² (2x + a) + bx + c for all x.

Solution

Find a and b if :

Problem 3 :

z⁴ + 4 = (z² + az + 2) (z² + bz + 2) for all z.

Solution

Problem 4 :

2z⁴ + 5z³ + 4z² + 7z + 6 = (z² + az + 2) (2z² + bz + 3) for all z.

Solution

Problem 5 :

Show that z⁴ + 64 can be factorised into two real quadratic factors of the form z² + az + 8 and z² + bz + 8, but cannot be factorised into two real quadratic factors of the form z² + az + 16 and z² + bz + 4.

Solution

Problem 6 :

Find real numbers a and b such that

x⁴ - 4x² - 8x - 4 = (x² + ax + 2) (x² + bx - 2)

and hence solve the equation. x⁴ + 8x = 4x² + 4.

Solution

Problem 1 :

2z – 3 is a factor of 2z³ – z² + az – 3. Find a and all zeros of the cubic.

Solution

Problem 2 :

3z + 2 is a factor of 3z³ – z² + [a + 1]z + a. Find a and all the zeros of the cubic.

Solution

Problem 3 :

Both 2x + 1 and x – 2 are factors of P(x) = 2x⁴ + ax³ + bx² – 12x – 8. Find a and b and all zeros of P(x).

Solution

Problem 4 :

x + 3 and 2x – 1 are factors of 2x⁴ + ax³ + bx² + ax + 3. Find a and b and hence determine all zeros of the quartic.

Solution

Problem 5 :

a. x³ + 3x²  - 9x + c has two identical linear factors. Prove that c is either 5 or -27 and factories the cubic into linear factors in each case.

Solution

Problem 6 :

3x³ + 4x² – x + m has two identical linear factors. Find m and find the zeros of the polynomial in all possible cases.

Solution