1.
Classify the polynomial by both degree and number of terms.-5x^{4} + 7x^{3}
Correct Answer
B. Quartic (degree 4) binomial
Explanation
The given polynomial is -5x^4 + 7x^3. It has a degree of 4 because the highest power of x is 4. It has two terms because there are two separate parts (-5x^4 and 7x^3) that are being added together. Therefore, the polynomial can be classified as a quartic (degree 4) binomial.
2.
Classify the polynomial by both degree and number of terms.4x^{5}
Correct Answer
D. Quintic (degree 5) monomial
Explanation
The given polynomial, 4x^5, is a monomial because it has only one term. Additionally, it is a quintic polynomial because the highest power of the variable, x, is 5. Therefore, the correct answer is "quintic (degree 5) monomial."
3.
Classify the polynomial by both degree and number of terms.2x^{3} + 3x^{2} - 5
Correct Answer
C. Cubic (degree 3) trinomial
Explanation
The given polynomial, 2x^3 + 3x^2 - 5, has a degree of 3 because the highest exponent in the polynomial is 3. It is also a trinomial because it has three terms, namely 2x^3, 3x^2, and -5. Therefore, the correct classification for this polynomial is cubic (degree 3) trinomial.
4.
Perform the indicated operation. Combine like terms and write your answer in standard form.(5x^{3} - 6x + 8) - (3x^{3} - 9)
Correct Answer
2x^3-6x+17
Explanation
The given expression is a subtraction of two polynomials. To simplify the expression, we need to combine like terms. Starting with the first polynomial, we have 5x^3 - 6x + 8. Then, subtracting the second polynomial, 3x^3 - 9, we can combine the like terms by subtracting the coefficients of the same degree of x. This gives us a final simplified expression of 2x^3 - 6x + 17.
5.
Perform the indicated operation. Combine like terms and write your answer in standard form.(3x^{4} - x + 3) + (3 - 2x - x^{4})
Correct Answer
C. 2x^4 - 3x + 6
Explanation
The given expression is the sum of two terms. The first term, 3x^4 - x + 3, has a degree of 4 and contains a variable x. The second term, 3 - 2x - x^4, also has a degree of 4 and contains a variable x. To combine like terms, we add the coefficients of the same degree terms. In this case, the x^4 term cancels out since it has opposite signs in both terms. The x term has a coefficient of -1 in the first term and -2 in the second term, so the sum is -1 - 2 = -3x. The constant term is 3 in the first term and 3 in the second term, so the sum is 3 + 3 = 6. Therefore, the correct answer is 2x^4 - 3x + 6.
6.
What is the degree of the polynomial 5x + 4x^{2} + 3x^{3} - 5x ?
Correct Answer
C. 3
Explanation
The degree of a polynomial is determined by the highest exponent of the variable. In this case, the polynomial is 3x^3 + 4x^2 + 5x - 5x. The highest exponent is 3, so the degree of the polynomial is 3.
7.
Perform the indicated operation. Write answer in standard form.(30x^{3} - 49x^{2} + 7x) + (50x^{3} - 75 - 60x^{2})
Correct Answer
80x^3-109x^2+7x-75
Explanation
The given expression involves adding two polynomials. To add polynomials, we combine like terms by adding their coefficients. In this case, we have two terms with x^3, two terms with x^2, and two terms with x. Adding the coefficients of each like term, we get 30x^3 + 50x^3 = 80x^3, -49x^2 - 60x^2 = -109x^2, and 7x - 75x = -68x. Therefore, the correct answer is 80x^3 - 109x^2 + 7x - 75.
8.
Classify by number of terms.x - 1
Correct Answer
B. Binomial
Explanation
The given expression "x - 1" has two terms, "x" and "-1". A binomial is a polynomial with two terms, so the expression "x - 1" is classified as a binomial.
9.
Classify by number of terms.4x^{2}
Correct Answer
A. Monomial
Explanation
The given expression, 4x^2, has only one term. In a polynomial expression, a term is a product of a coefficient and one or more variables raised to a power. Since there is only one term in 4x^2, it is classified as a monomial.
10.
Classify by number of terms.8x^{2} + 3x - 1
Correct Answer
C. Trinomial
Explanation
The given expression, 8x^2 + 3x - 1, has three terms: 8x^2, 3x, and -1. Therefore, it is classified as a trinomial.
11.
Classify by degree.7x^{4} + x^{2} - 2
Correct Answer
E. Quartic (degree 4)
Explanation
The given expression is a quartic polynomial because the highest power of the variable x is 4. A quartic polynomial is a polynomial of degree 4, which means it has terms with powers ranging from 0 to 4. In this case, the expression has a term with x raised to the power of 4, making it a quartic polynomial.
12.
Classify by degree.x^{2} - 1
Correct Answer
C. Quadratic (degree 2)
Explanation
The given expression x^2 - 1 is a quadratic polynomial because it is of degree 2. In a quadratic polynomial, the highest power of the variable is 2. The term x^2 represents the quadratic term, and -1 represents the constant term. Hence, the expression x^2 - 1 is a quadratic polynomial of degree 2.
13.
Classify by degree.10
Correct Answer
A. Constant (degree 0)
Explanation
The given answer is "constant (degree 0)" because the term "classify by degree" refers to categorizing functions based on the highest power of the variable in the function. In this case, since there is no variable present in the given statement, it can be classified as a constant function. A constant function has a degree of 0 because there are no variables raised to any power in the function.
14.
Classify by degree.-22x
Correct Answer
B. Linear (degree 1)
Explanation
The given expression, -22x, is a linear function because it only contains the variable x raised to the first power. In other words, there are no x terms raised to any other powers, such as x^2, x^3, etc. Therefore, the degree of this expression is 1, making it a linear function.
15.
Perform the indicated operation, then write answer in standard form.(2d^{2} + 9) - (3d^{2} - 7)
Correct Answer
-d^2+16
-1d^2+16
Explanation
The given expression requires subtracting the second polynomial from the first. To do this, we subtract the corresponding terms. In this case, we have 2d^2 - 3d^2 = -d^2, and 9 - (-7) = 16. Therefore, the resulting polynomial is -d^2 + 16. Another way to represent the same polynomial is -1d^2 + 16.
16.
Write in standard form: x^{2} - x^{4} + 2x^{2}
Correct Answer
-x^4+3x^2
-1x^4+3x^2
Explanation
The given expression is x^2 - x^4 + 2x^2. Simplifying this expression, we can combine like terms to get -x^4 + 3x^2. Therefore, the correct answer is -x^4 + 3x^2. Another way to represent this expression is -1x^4 + 3x^2.
17.
Write in standard form: 2m^{2} - 3 + 7m
Correct Answer
2m^2+7m-3
Explanation
The given expression is in standard form, which means it is written with the terms arranged in descending order of their exponents. In this case, the terms are already arranged in descending order of their exponents: 2m^2, 7m, -3. Therefore, the given expression 2m^2 + 7m - 3 is already in standard form.