➕ Algebra 1 · Exponents and Polynomials

Adding and Subtracting Polynomials: true or false

Decide fast, then read the reason. Wrong answers are the best teachers, so every statement explains itself.

All statements and answers

• 7x⁻² + 1 is a polynomial.

  ✗ False. The exponents in a polynomial must be whole numbers. The term 7x⁻² = 7/x² has a negative exponent.

• x² − 9 is a binomial.

  ✓ True. It has two terms, x² and −9.

• The degree of 4x² − 9x⁵ + 2 is 2.

  ✗ False. With one variable, the degree is the highest exponent, which is 5 (from −9x⁵). The first term written is not always the highest.

• (3x + 7) − (x + 4) = 2x + 3.

  ✓ True. Flip both signs of the second group: 3x + 7 − x − 4. Then 3x − x = 2x and 7 − 4 = 3.

• (8x + 5) − (3x − 2) = 5x + 3.

  ✗ False. The −2 flips to +2: 8x + 5 − 3x + 2 = 5x + 7. Check with x = 3: the original is 29 − 7 = 22, and 5(3) + 7 = 22, but 5(3) + 3 = 18.

• The sum of two polynomials is always a polynomial.

  ✓ True. Adding combines like terms and never changes the exponents, so the exponents stay whole numbers. Polynomials are closed under addition.

• Adding two polynomials of degree 3 always gives a polynomial of degree 3.

  ✗ False. Leading terms can cancel: (x³ + 2x) + (−x³ + 5) = 2x + 5, which has degree 1.

• 6x² + 4x² = 10x⁴.

  ✗ False. Adding like terms adds the coefficients and keeps the exponent: 6x² + 4x² = 10x².

• x = 0 is the best value to test, because the arithmetic is easy.

  ✗ False. At x = 0 every term with an x becomes 0, so only the constants get tested. A mistake in the x terms would go unnoticed. Use a value like x = 3.

• (x² + 4x) + (3x² − 4x) = 4x².

  ✓ True. The x² terms give 1 + 3 = 4, and the x terms cancel: 4x − 4x = 0.

• −(5x − 3) = −5x − 3.

  ✗ False. The minus sign flips both terms: −(5x − 3) = −5x + 3.

• 8 − x + 2x² is written in standard form.

  ✗ False. Standard form lists the terms from the highest power to the lowest. The standard form is 2x² − x + 8.

• The number 5 is a polynomial.

  ✓ True. A nonzero constant is a monomial of degree 0, so it is a polynomial.

• Subtracting a polynomial gives the same result as adding its opposite.

  ✓ True. A − B = A + (−B), where −B is B with the sign of every term changed.