Question

It is given a polynomial f(x) = 2x³ + ax² - bx + 12, where a and b are constants. When f(x) is divided by x-2, the remainder is -18. It is known that f(x) is divisible by x + 4. a) Find the values of a and b b) Solve the equation f(x) = 0​

Answer 1

Answer:

Values of constant a and b:

a= 1/2

b= 2

2x³ + ax² - bx + 12=0; x  ≠ 0

Step-by-step explanation:

What is a Polynomial Equation?

The equations formed with variables, exponents and coefficients are called as polynomial equations. It can have different exponents, where the higher one is called the degree of the equation. We can solve polynomials by factoring them in terms of degree and variables present in the equation.

A polynomial function is an expression which consists of a single independent variable, where the variable can occur in the equation more than one time with different degree of the exponent. Students will also learn here how to solve these polynomial functions. The graph of a polynomial function can also be drawn using turning points, intercepts, end behaviour and the Intermediate Value Theorem.

Example of polynomial function:

f(x) = 3x2 + 5x + 19

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Answer 2

Answer:

Step-by-step explanation:

(a). 2( 2 )³ + a ( 2 )² - b ( 2 ) + 12 + ( 18 ) = 0

16 + 4a - 2b + 30 = 0

4a - 2b = - 46

2a - b = - 23 ....... ( 1 )

2 ( - 4 )³ + a ( - 4 )² - b ( - 4 ) + 12 = 0

- 128 + 16a + 4b + 12 = 0

16a + 4b = 116

4a + b = 29 ........ ( 2 )

( 1 ) + ( 2 )

6a = 6 , a = 1

4(1) + b = 29 , b = 25

(b). 2x³ + x² + 25x + 12 = 0

$x_{1}$ = - 4 ( given: f(x) is divisible by x + 4 )

$x_{2}$ = 0.5

$x_{3}$ = 3