Test: Polynomials Basics - ACT MCQ

A polynomial of degree 2 is called quadratic polynomial.
Quadratic polynomials are of the form ax² + bx + c and it can contain at most three terms namely ax², bx and c. Thus, we can say that a quadratic polynomial can have at most three terms.
Similarly, a polynomial of degree 1 is called linear polynomial and a polynomial of degree 3 is called cubic polynomial.

If the discriminant of a quadratic polynomial, D > 0, then the polynomial has two real and unequal roots.

Degree of the zero polynomial is not defined.
Zero polynomial is denoted by 0, and degree for that is not defined.

S is the sum of zeroes and P is the product of zeroes: 

S = (3 + √2) + (3 – √2) = 6

P = (3 + √2) x (3 – √2) = (3)² – (√2)² = 9 – 2 = 7

So, Quadratic polynomial = x² – Sx + P = x² – 6x + 7

Degree of a non-zero constant polynomial is zero.
We can see that given polynomial 7 contain only one term and that is constant. 7 can also be written as 7x⁰.
Hence degree of 7 is zero.

x² - 27 = 0

x² = 27

x = √27

x = ±3√3

Degree of a polynomial is the highest power of variable in a polynomial.
The term with the highest power of x is 4x⁷ and exponent of x in that term is 7, so the degree of polynomial of 4x⁷ + 9x⁵ + 5x² + 11 is 7.

Quadratic polynomial is x² – Sx + P = 0, where S is the sum and P is the product

⇒ x² – (-6)x + 5 = 0

⇒ x² + 6x + 5 = 0

Coefficient is the number which is multiplied with respective variable.
In the given polynomial 6x⁴ + 3x² + 8x + 5, there is not an expression containing x³. So we can write 6x⁴ + 3x² + 8x + 5 as 6x⁴ + 0x³ + 3x² + 8x + 5. We can see that 0 is multiplied with expression x³, so coefficient of x³ is 0.

For an expression to be a polynomial, exponent of variable has to be whole number.
 can be written as x^-1/2. We can see that exponent of x is -1/2 which is not whole number (W = {0, 1, 2, 3…}). Hence, 1x√2 is not a polynomial.

A polynomial's maximum number of zeroes equals the polynomial's degree.

The number of zeroes of a polynomial is equal to the number of points where the graph of polynomial intersects the x-axis.

Let p(x) = mx + n

Put x = y

p(y) = my + n = 0

So, y is zero of p(x).

In every polynomial, the highest power of the variable is called a degree.

Discriminant will be equal to zero for equal roots:

b² - 4ac = 0

b² = 4ac

ac = b²/4

ac > 0 (square of any number cannot be negative)