WorksheetsExploring Polynomial Zeros and Multiplicities
Total questions: 23
Worksheet time: 12mins
What is a zero of a polynomial function?
A zero of a polynomial function is a value that makes the polynomial equal to zero.
A zero of a polynomial function is a constant term in the polynomial.
A zero of a polynomial function is a value that makes the polynomial positive.
A zero of a polynomial function is the highest degree term.
Which of the following is a zero of the polynomial f(x) = x^2 - 4?
3
-1
2, -2
1
If a polynomial has a zero at x = 3 with multiplicity 2, what does this imply about the graph at x = 3?
The graph is flat at x = 3.
The graph crosses the x-axis at x = 3.
The graph has a hole at x = 3.
The graph touches the x-axis at x = 3 and does not cross it.
How many times does the zero x = -1 occur in the polynomial f(x) = (x + 1)^3(x - 2)?
4
2
1
3
What is the multiplicity of the zero x = 2 in the polynomial f(x) = (x - 2)^4?
4
5
3
2
Which of the following polynomials has a zero of multiplicity 1?
(x - 2)
(x + 3)
(x^2 - 4)
(x - 2)(x + 1)
What is the end behavior of the polynomial f(x) = -2x^3 + 5x - 1 as x approaches positive infinity?
f(x) approaches positive infinity as x approaches positive infinity.
f(x) approaches zero as x approaches positive infinity.
f(x) approaches negative infinity as x approaches positive infinity.
f(x) oscillates between positive and negative values as x approaches positive infinity.
How can you determine the multiplicity of a zero from the factored form of a polynomial?
The multiplicity of a zero is the number of times it appears in the polynomial's standard form.
The multiplicity of a zero is always equal to one for any polynomial.
The multiplicity of a zero can be found by taking the derivative of the polynomial.
The multiplicity of a zero is the exponent of its corresponding factor in the polynomial's factored form.
Which of the following is the correct factorization of the polynomial f(x) = x^2 - 5x + 6?
(x - 4)(x - 2)
(x + 2)(x + 3)
(x - 1)(x - 6)
(x - 2)(x - 3)
What is the end behavior of the polynomial f(x) = 3x^4 - 2x^2 + 1 as x approaches negative infinity?
f(x) approaches a constant value as x approaches negative infinity.
f(x) approaches negative infinity as x approaches negative infinity.
f(x) approaches positive infinity as x approaches negative infinity.
f(x) approaches zero as x approaches negative infinity.
If a polynomial has a zero at x = 0 with multiplicity 3, how does the graph behave at this point?
The graph crosses the x-axis at x = 0.
The graph has a vertical asymptote at x = 0.
The graph forms a sharp corner at x = 0.
The graph touches the x-axis and flattens out at x = 0.
Which of the following is a correct statement about the zeros of the polynomial f(x) = x^4 - 16?
The polynomial has two zeros: 4 and -4.
The polynomial has four zeros: 2, -2, 2i, and -2i.
The polynomial has three zeros: 2, -2, and 0.
The polynomial has five zeros: 2, -2, 2i, -2i, and 1.
What is the multiplicity of the zero x = 1 in the polynomial f(x) = (x - 1)^2(x + 3)?
1
3
4
2
How do you find the zeros of a polynomial function?
Differentiate the polynomial and set it to zero.
Set the polynomial equal to zero and solve for the variable.
Graph the polynomial to find the zeros.
Use the quadratic formula for all polynomial functions.
What is the significance of the leading coefficient in determining the end behavior of a polynomial?
The leading coefficient has no impact on the polynomial's graph.
The leading coefficient determines the number of roots of the polynomial.
The leading coefficient indicates the direction of the polynomial's end behavior.
The leading coefficient affects the polynomial's degree only.
Which of the following polynomials has a zero at x = 5 with multiplicity 3?
(x - 5)^2
(x - 5)^3
(x - 5)^4
(x + 5)^3
What does it mean for a polynomial to have an even degree?
A polynomial has an even degree if it can be factored into linear terms.
A polynomial has an even degree if its coefficients are all positive.
A polynomial has an even degree if it has no real roots.
A polynomial has an even degree if its highest exponent is an even number.
How can you tell if a polynomial has complex zeros?
A polynomial has complex zeros if it has real coefficients only.
A polynomial has complex zeros if its discriminant is negative.
A polynomial has complex zeros if it has an even degree.
A polynomial has complex zeros if its leading coefficient is positive.
What is the relationship between the degree of a polynomial and the number of zeros it can have?
A polynomial of degree n can have infinitely many zeros.
A polynomial of degree n can have at most n zeros.
A polynomial of degree n can have no zeros.
A polynomial of degree n has exactly n zeros.
Which of the following statements is true regarding the graph of a polynomial function at a zero with odd multiplicity?
The graph crosses the x-axis at the zero.
The graph touches the x-axis at the zero.
The graph does not intersect the x-axis at the zero.
The graph has a local maximum at the zero.
What are the zeros and y-intercept of the function?
x-intercepts = 4,1
y-intercept = 3
x-intercepts = 3,1,-2,-5
y-intercept = 3.5
x-intercepts = -3, -1, 2, 5
y-intercept = 3
x-intercepts = -2,4
x-intercepts = -7
What is the degree of the polynomial shown in the image?
4
5
7
6
