Introduction To Polynomials - Free MCQ Practice Test with solutions,

To solve the quadratic equation x² + 7x + 12 = 0, we use the factorization method.

• We need to find two numbers that multiply to give 12 (the constant term) and add up to give 7 (the coefficient of x). The numbers 3 and 4 satisfy this because 3 * 4 = 12 and 3 + 4 = 7.

• Therefore, we can factor the quadratic equation as:
  x² + (4+3)x + 12 = 0
  x² + 4x + 3x + 12 = 0
  x(x + 4) + 3(x + 4) = 0
  (x + 3)(x + 4) = 0

• Setting each factor equal to zero:
  x + 3 = 0 → x = -3
  x + 4 = 0 → x = -4

To calculate the expression involving cubes, let's break it down step by step:

• 3 cubed: 3³ = 27
• 5 cubed: 5³ = 125
• 8 cubed: 8³ = 512

Now, we can substitute these values into the expression:

3³ + 5³ - 8³ = 27 + 125 - 512

Calculating this step-by-step:

• First, add 27 and 125: 27 + 125 = 152
• Then, subtract 512: 152 - 512 = -360

Therefore, the final result is: -360.

⇒ x²+ 5x - 6 = 0

⇒ x²+ 6x -x - 6 = 0

⇒ x(x+6) -1(x+6) = 0

⇒ (x-1) (x+6) = 0

⇒ x = 1, -6

Therefore, the option that is not a quadratic polynomial is: x – x³

p(x) = ax
p(x) = 0
ax = 0
x = 0/a
x = 0
{0 by something is equals to 0}
Therefore,0 is the zero of the polynomial. 
Checking:-
p(0) = a(0)
= 0

- To find p(-2), substitute -2 into the function p(x) = 7 - 3x + 2x^2.
- Calculate step-by-step:
- p(-2) = 7 - 3(-2) + 2(-2)²
- This simplifies to 7 + 6 + 2(4)
- Which further simplifies to 7 + 6 + 8
- Adding these together gives: 21.
- Therefore, the value of p(-2) is 21, making option 3 the correct answer.

A linear polynomial has 1 zero.

A quadratic polynomial has 2 zeroes.

A cubic polynomial has 3 zeroes.

In general, any polynomial has as many zeroes as its degree.

A cubic polynomial is a polynomial of degree 3. A univariate cubic polynomial has the form An equation involving a cubic polynomial is called a cubic equation. A closed-form solution known as the cubic formula exists for the solutions of an arbitrary cubic equation.

To find the value of the polynomial ( 6 - 4x + 3x² ) at ( x = 3 ):

- Substitute ( x = 3 ) into the polynomial:
=[6 - 4(3) + 3(3²)]
- Calculate step-by-step:
=[6 - 4(3) + 3(9)]
=[6 - 12 + 27]
=[21]

Thus, the value of the polynomial at ( x = 3 ) is 21.

Therefore, the correct answer is:
- B: 21

A zero polynomial is a polynomial in which all the coefficients are 0.

Let the polynomials highest power variable be x^n as per the above statement x^n = 0 

Now, The degree of the zero polynomial is log0 which is undefined. 

Hence. The degree of zero polynomial is undefined.

Zero of the zero polynomial is any real number.
e.g., Let us consider zero polynomial be 0(x-k), where k is a real number For determining the zero, put x-k = 0 ⇒ x = k Hence, zero of the zero polynomial be any real number. 

The degree refers to the highest power of the polynomial. In this polynomial X has highest power 4.So the degree of polynomial is 4.