Question 1

The zero of the polynomial $p(x) = 3x - 2$ is:

Accepted Answer

To find the zero, set $p(x) = 0$: $$3x - 2 = 0 \implies 3x = 2 \implies x = \frac{2}{3}$$

Question 2

If $x + y + z = 0$, then $x^3 + y^3 + z^3$ is equal to:

Accepted Answer

Using the algebraic identity $x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)$,
when $x + y + z = 0$, the right side becomes $0$. Therefore, $x^3 + y^3 + z^3 = 3xyz$.

Question 3

What is the degree of the polynomial $p(x) = x^5 - x^4 + 3$?

Accepted Answer

The degree of a polynomial is the highest power of the variable $x$. In $p(x) = x^5 - x^4 + 3$, the highest power of $x$ is $5$.

Question 4

The remainder when $x^3 + 1$ is divided by $x + 1$ is:

Accepted Answer

By Remainder Theorem, the remainder when $p(x) = x^3 + 1$ is divided by $x + 1$ (i.e. $x - (-1)$) is $p(-1)$. $$p(-1) = (-1)^3 + 1 = -1 + 1 = 0$$

Question 5

The factors of $x^2 - 9$ are:

Accepted Answer

Using identity $a^2 - b^2 = (a - b)(a + b)$: $$x^2 - 9 = x^2 - 3^2 = (x - 3)(x + 3)$$

Polynomials