Practice
Adding, Subtracting, Multiplying, and Dividing Polynomials Practice
Fifty original questions covering structure, degree, operations, long division, missing terms, and error analysis.
- Answered
- 0 / 50
- Correct
- 0
- Incorrect
- 0
- Accuracy
- 0%
Question 1
Explanation
Terms are separated by addition or subtraction. This expression has one nonzero term, so it is a monomial.
Question 2
Explanation
Terms are separated by addition or subtraction. This expression has two nonzero terms, so it is a binomial.
Question 3
Explanation
Terms are separated by addition or subtraction. This expression has three nonzero terms, so it is a trinomial.
Question 4
Explanation
Terms are separated by addition or subtraction. This expression has four nonzero terms, so it is a polynomial with four terms.
Question 5
Explanation
Terms are separated by addition or subtraction. This expression has one nonzero term, so it is a monomial.
Question 6
Explanation
Add the exponents of all variable factors. Their sum is \(5\), so the monomial has degree \(5\).
Question 7
Explanation
Add the exponents of all variable factors. Their sum is \(8\), so the monomial has degree \(8\).
Question 8
Explanation
Add the exponents of all variable factors. Their sum is \(1\), so the monomial has degree \(1\).
Question 9
Explanation
Add the exponents of all variable factors. Their sum is \(7\), so the monomial has degree \(7\).
Question 10
Explanation
Add the exponents of all variable factors. Their sum is \(9\), so the monomial has degree \(9\).
Question 11
Explanation
Find each term's degree, then select the greatest. The largest term degree is \(6\).
Question 12
Explanation
Find each term's degree, then select the greatest. The largest term degree is \(5\).
Question 13
Explanation
Find each term's degree, then select the greatest. The largest term degree is \(4\).
Question 14
Explanation
Find each term's degree, then select the greatest. The largest term degree is \(4\).
Question 15
Explanation
Find each term's degree, then select the greatest. The largest term degree is \(5\).
Question 16
Explanation
Like terms have exactly the same variables raised to exactly the same powers. \(6x^2y\) and \(-9x^2y\) meet that test.
Question 17
Explanation
Like terms have exactly the same variables raised to exactly the same powers. \(4ab^3\) and \(11ab^3\) meet that test.
Question 18
Explanation
Like terms have exactly the same variables raised to exactly the same powers. \(-5m^2n^4\) and \(2m^2n^4\) meet that test.
Question 19
Explanation
Like terms have exactly the same variables raised to exactly the same powers. \(7p\) and \(-3p\) meet that test.
Question 20
Explanation
Like terms have exactly the same variables raised to exactly the same powers. \(8r^3s^2\) and \(r^3s^2\) meet that test.
Question 21
Explanation
Align equal powers and add their coefficients. The coefficient-by-coefficient sum gives \(5x^{2}+5x-2\).
Question 22
Explanation
Align equal powers and add their coefficients. The coefficient-by-coefficient sum gives \(2x^{3}+x^{2}+5x+6\).
Question 23
Explanation
Align equal powers and add their coefficients. The coefficient-by-coefficient sum gives \(4x^{3}+2x^{2}-x+3\).
Question 24
Explanation
Align equal powers and add their coefficients. The coefficient-by-coefficient sum gives \(-3x^{2}-5x+5\).
Question 25
Explanation
Align equal powers and add their coefficients. The coefficient-by-coefficient sum gives \(5x^{3}+5x^{2}+3x+3\).
Question 26
Explanation
Distribute the subtraction sign to every term in the second polynomial, then combine like terms. The result is \(-5x^{2}-4x+10\).
Question 27
Explanation
Distribute the subtraction sign to every term in the second polynomial, then combine like terms. The result is \(-6x^{3}+3x^{2}+x-3\).
Question 28
Explanation
Distribute the subtraction sign to every term in the second polynomial, then combine like terms. The result is \(5x^{2}+12x-10\).
Question 29
Explanation
Distribute the subtraction sign to every term in the second polynomial, then combine like terms. The result is \(x^{3}-5x^{2}-10x+10\).
Question 30
Explanation
Distribute the subtraction sign to every term in the second polynomial, then combine like terms. The result is \(9x^{3}-9x^{2}+10x-3\).
Question 31
Explanation
Distribute \(3x^{2}\) to every term, multiplying coefficients and adding variable exponents. This gives \(15x^{4}-6x^{3}+12x^{2}\).
Question 32
Explanation
Distribute \(-2x\) to every term, multiplying coefficients and adding variable exponents. This gives \(8x^{3}-6x\).
Question 33
Explanation
Distribute \(5x^{3}\) to every term, multiplying coefficients and adding variable exponents. This gives \(10x^{5}+30x^{4}-5x^{3}\).
Question 34
Explanation
Distribute \(-4x^{2}\) to every term, multiplying coefficients and adding variable exponents. This gives \(-4x^{5}+12x^{3}-8x^{2}\).
Question 35
Explanation
Distribute \(6x\) to every term, multiplying coefficients and adding variable exponents. This gives \(-12x^{3}+24x^{2}-30x\).
Question 36
Explanation
Divide every term by \(3x^2\), dividing coefficients and subtracting exponents. The quotient is \(6x^4-3x^2+4\).
Question 37
Explanation
Divide every term by \(-2y\), dividing coefficients and subtracting exponents. The quotient is \(-4y^3+3y^2-5\).
Question 38
Explanation
Divide every term by \(5a^3\), dividing coefficients and subtracting exponents. The quotient is \(5a^4+3a^2-2\).
Question 39
Explanation
Divide every term by \(4m^2n\), dividing coefficients and subtracting exponents. The quotient is \(5m^3n^2-3mn+2\).
Question 40
Explanation
Divide every term by \(-6p^2\), dividing coefficients and subtracting exponents. The quotient is \(-5p^4+3p^2-2\).
Question 41
Explanation
Long division produces quotient \(2x^{2}-x+3\) and remainder \(4\). Multiplication verifies \(2x^{3}-5x^{2}+5x-2=(x-2)(2x^{2}-x+3)+4\), and the constant remainder has degree below the linear divisor.
Question 42
Explanation
Long division produces quotient \(x^{2}+4x-2\) and remainder \(-5\). Multiplication verifies \(x^{3}+7x^{2}+10x-11=(x+3)(x^{2}+4x-2)-5\), and the constant remainder has degree below the linear divisor.
Question 43
Explanation
Long division produces quotient \(x^{3}-2x^{2}+5\) and remainder \(3\). Multiplication verifies \(x^{4}-3x^{3}+2x^{2}+5x-2=(x-1)(x^{3}-2x^{2}+5)+3\), and the constant remainder has degree below the linear divisor.
Question 44
Explanation
Long division produces quotient \(2x^{3}-3x+1\) and remainder \(6\). Multiplication verifies \(2x^{4}+4x^{3}-3x^{2}-5x+8=(x+2)(2x^{3}-3x+1)+6\), and the constant remainder has degree below the linear divisor.
Question 45
Explanation
Long division produces quotient \(3x^{2}+2x-1\) and remainder \(-7\). Multiplication verifies \(3x^{3}-10x^{2}-9x-3=(x-4)(3x^{2}+2x-1)-7\), and the constant remainder has degree below the linear divisor.
Question 46
Explanation
The correct analysis is: The degree is 7 because the first term has total degree 7. Polynomial structure and the identity \(P=DQ+R\) determine the result.
Question 47
Explanation
The correct analysis is: The negative sign was not distributed to the 5x term. Polynomial structure and the identity \(P=DQ+R\) determine the result.
Question 48
Explanation
The correct analysis is: It preserves alignment of equal powers during subtraction. Polynomial structure and the identity \(P=DQ+R\) determine the result.
Question 49
Explanation
The correct analysis is: Its degree is less than the divisor's degree. Polynomial structure and the identity \(P=DQ+R\) determine the result.
Question 50
Explanation
The correct analysis is: Expand (x-2)Q and add 5 to recover P. Polynomial structure and the identity \(P=DQ+R\) determine the result.
Keyboard: use Tab to move, arrow keys to change answer choices, and Enter to check an answer.
Your practice summary
Use the results to decide what to review before your next attempt.
- Correct
- 0
- Incorrect
- 0
- Completed
- 50 / 50
Questions to review
No mistakes this time. Excellent work.
- Question 1Polynomial classificationEasy
- Question 2Polynomial classificationEasy
- Question 3Polynomial classificationEasy
- Question 4Polynomial classificationEasy
- Question 5Polynomial classificationEasy
- Question 6Degree of a monomialEasy
- Question 7Degree of a monomialEasy
- Question 8Degree of a monomialEasy
- Question 9Degree of a monomialEasy
- Question 10Degree of a monomialEasy
- Question 11Degree of a polynomialEasy
- Question 12Degree of a polynomialEasy
- Question 13Degree of a polynomialEasy
- Question 14Degree of a polynomialEasy
- Question 15Degree of a polynomialEasy
- Question 16Identify like termsMedium
- Question 17Identify like termsMedium
- Question 18Identify like termsMedium
- Question 19Identify like termsMedium
- Question 20Identify like termsMedium
- Question 21Add polynomialsMedium
- Question 22Add polynomialsMedium
- Question 23Add polynomialsMedium
- Question 24Add polynomialsMedium
- Question 25Add polynomialsMedium
- Question 26Subtract polynomialsMedium
- Question 27Subtract polynomialsMedium
- Question 28Subtract polynomialsMedium
- Question 29Subtract polynomialsMedium
- Question 30Subtract polynomialsMedium
- Question 31Multiply a polynomial by a monomialMedium
- Question 32Multiply a polynomial by a monomialMedium
- Question 33Multiply a polynomial by a monomialMedium
- Question 34Multiply a polynomial by a monomialMedium
- Question 35Multiply a polynomial by a monomialMedium
- Question 36Divide a polynomial by a monomialMedium
- Question 37Divide a polynomial by a monomialMedium
- Question 38Divide a polynomial by a monomialMedium
- Question 39Divide a polynomial by a monomialMedium
- Question 40Divide a polynomial by a monomialMedium
- Question 41Polynomial long divisionHard
- Question 42Polynomial long divisionHard
- Question 43Polynomial long divisionHard
- Question 44Polynomial long divisionHard
- Question 45Polynomial long divisionHard
- Question 46Polynomial degree error analysisHard
- Question 47Subtraction error analysisHard
- Question 48Missing-degree termsHard
- Question 49Remainder interpretationHard
- Question 50Division verificationHard