Login or Sign Up

Ask a New Question

i) Given that x^4-2x^3+ax^2-2 =x(x-1)(x+1)Q(x)+4x^2+bx+c is an identity, state the degree of the polynomial Q(x).

ii) Find the value of a, of b and of c. Hence, state the remainder when x^4-2x^3+ax^2-2 is divided by x^3-x

Step by step please🙏Thank you

You and "Dan" need to get together. See

www.jiskha.com/questions/1894782

Did you know?

Did you know that in algebra, polynomial identities can help us find the degree and values of unknown polynomials? In this example, we have the polynomial identity x^4-2x^3+ax^2-2 = x(x-1)(x+1)Q(x)+4x^2+bx+c. To determine the degree of the polynomial Q(x), we can examine the left-hand side of the equation. The highest power of x is x^4, which means Q(x) must have a degree of at least 1 less than x^4, so the degree of Q(x) is 3. Now, let's find the values of a, b, and c. We can accomplish this by comparing the coefficients of like terms on both sides of the equation. Starting with x^3, there is no x^3 term on the right-hand side, so the coefficient of x^3 on the left-hand side is -2. Since this comes from the term -2x^3, we have -2 = -2, giving us no information about a. Moving on to x^2, we see that -2 is the constant term on the right-hand side, so the coefficient of x^2 on the left-hand side must also be -2. This gives us ax^2 = -2, allowing us to determine that a = -2. Next, we look at the x term. On the left-hand side, the coefficient is 0, and on the right-hand side, there is no x term. The equation 0 = 0 does not provide any information about b. Finally, focusing on the constant term, we see that -2 is the constant term on the left-hand side, so c must also be -2. We can conclude that a = -2, b is undetermined, and c = -2. Interestingly, by using the information we gathered about a, b, and c, we can determine the remainder when x^4-2x^3+ax^2-2 is divided by x^3-x. This remainder is simply 0.