Polynomials

Question

Answer 1

The polynomial f(x) = 2x³ - 5x² + ax + b

1 as a root.

So the remainder is zero.

f(1) = 2(1)³ - 5(1)² + a(1) + b

0 = 2 - 5 + a + b

- 2 + 5 = a + b

a + b = 3 ---> (1)

 

From remainder theorem,

When you divide a polynomial f(x) by x - a the remainder r will be f(a).

The remainder 16 when divided by x -  4.

f(4) = 2(4)³ - 5(4)² + a(4) + b

16 = 128 - 80 + 4a + b

16 - 128 + 80 = 4a + b

4a + b = - 32 ---> (2)

 

Now solve the equations (1) and (2) in Gauss reduction method.

Write the equations into matrices form AX = B

A is coefficient matrix, X is variable matrix and B is constant matrix.

The augmented matrix is

are represents first row, second row.

 

Substitute the a, b values in polynomial function f(x) = 2x³ - 5x² + ax + b

f(x) = 2x³ - 5x² - (35/3)x + (44/3).