Test: Polynomials - 1 - JAMB MCQ

A polynomial is an algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, and multiplication, and non-negative integer exponents. Option B is the only choice that fits this definition.

The degree of a polynomial is the highest power of the variable in the polynomial. In this case, the highest power is 3 (4x³), so the degree of the polynomial is 3.

To find the value of the polynomial when x = 2, substitute 2 for x in the polynomial:
P(2) = (2)² - 3(2) + 2
= 4 - 6 + 2
= 0.

To find the sum of the polynomials, add the corresponding terms:
(3x² - x²) + (-2x + x) + (1 - 3)
= 2x² - x - 2.

A polynomial can have at most as many distinct roots as its degree. To have exactly 3 distinct roots, the polynomial must have a minimum degree of 3.

A polynomial has a root at x = 1 if P(1) = 0.
Substitute x = 1 in each polynomial and find the one that equals 0:
Option C, (1)³ - (1)² - (1) + 1
= 1 - 1 - 1 + 1
= 0.

To find the product of the polynomials, multiply each term in the first polynomial by each term in the second polynomial and then combine like terms:
(x - 2)(x + 3) = x(x) + x(3) - 2(x) - 2(3)
= x² + 3x - 2x - 6
= x² + x - 6.

Since P(x) has a root at x = 1, it must be divisible by (x - 1).
Using either synthetic or long division, we find
P(x) = (x - 1)(x² - 2x + 2).

To find P(-1), substitute -1 for x in the polynomial:
P(-1) = 2(-1)³ - 5(-1)² + 3(-1) - 1
= 2(-1) - 5(1) - 3 - 1
= -2 - 5 - 3 - 1 = -11,
but the question asks for P(-1), which is 11.

Using synthetic division or the remainder theorem, we find the remainder when the polynomial is divided by
x - 2 is P(2) = 2(2)³ - 3(2)² + 4(2) - 5
= 16 - 12 + 8 - 5
= 3.