Test: Polynomial Zeros - ACT MCQ

Since, the graph is completely above or below the x-axis, hence, it has no real roots. If a polynomial has real roots only then it cuts the x-axis. If it lies above or below, the roots are complex in nature.

An expression in the form of (x) = a₀ + a₁x + a₂x² + … + a_nxⁿ, where a_n ≠ 0, is called a polynomial where a₁, a₂ … a_n are real numbers and each power of x is a non-negative integer.
In case of √x + 2x + 4, the power of x is not an integer.
Therefore it is not a polynomial.

An expression in the form of (x) = a₀ + a₁x + a₂x² + … + a_nxⁿ, where a_n ≠ 0, is called a polynomial where a₁, a₂ … a_n are real numbers and each power of x is a non-negative integer.
In case of √x + 2x + 4 , the power of √x is not an integer. Similarly for x^2/3 + 10x, 2/3 is a fraction.
Now, 5x + 5/x in this case the power of x is a negative integer. Hence it is not a polynomial.

The leading coefficient of the polynomial is less than zero, hence, it has downward opening. For example, the graph of -x² is

If the graph meets x-axis at one point only, then the quadratic polynomial has coincident zeros. Also, the discriminant of the quadratic polynomial is zero, therefore roots will be real.

Since, the graph of the polynomial cuts the x-axis at 3 points, hence, it will be a cubic polynomial. A polynomial is said to be linear, quadratic, cubic or biquadratic according to the degree of the polynomial.

The graph of x² + 2x + 1 does not cut the x-axis, because it has imaginary roots.
x² + 2x + 1 = 0
x² + x + x + 1 = 0
x(x + 1) + (x + 1) = 0
(x + 1)(x + 1) = 0
x = -1, -1

The graph of the polynomial cuts the x-axis. Only the zeros of the polynomial cut the x-axis.
4x² - 8x + 3 = 0
4x² - 6x - 2x + 3 = 0
2x(2x - 3) - 1(2x - 3) = 0
(2x - 3)(2x - 1) = 0

Hence, the graph of the polynomial cuts the x-axis at 

Since, the zeros of the polynomial are 3 and -5.
Therefore, x = 3 and x = -5 and they cut the x-axis so the y-coordinate will be zero.
Hence, the points it cuts the x-axis will be (3, 0) and (-5, 0).

A biquadratic polynomial has highest power 4.
Hence, the polynomial with the highest power as 4 is x⁴ - 9 or (x² + 3)(x² - 3).

The degree of the polynomial is the highest of the degree of the polynomial. Hence, a polynomial with highest degree one is linear, two as quadratic and so on.

A number is called zero of polynomial when it satisfies the equation of the polynomial.

The zeros of the quadratic polynomial cut the x-axis at two different points.
∴ b² – 4ac ≥ 0
Hence, the nature of the zeros will be real and distinct.

The graph of the polynomial will have a downward opening since, a < 0
The graph for the same can be observed here,

The graph of the polynomial 2x² - 8x + 5 cuts the y-axis.
Hence, the value of x will be 0.
y(0) = 2(0)² - 8(0) + 5
y = 5
The graph cuts the y-axis at (0, 5)