Operations with Polynomials 4.1

Open Ended

Why is understanding the properties of exponents important when working with polynomials?

Multiple Choice

Which property of exponents allows you to multiply two powers with the same base by adding their exponents?

• Product of Powers

• Quotient of Powers

• Power of a Power

• Negative Exponent

Multiple Choice

Which of the following is NOT a requirement for a monomial expression to be in simplified form?

• There are no powers of powers.

• Each base appears exactly once.

• All fractions are in simplest form.

• There are positive exponents.

Open Ended

Explain the steps involved in simplifying the expression (2a^-2)(3a^3b^2)(c^-2).

Fill in the Blanks

Fill in the blank: To simplify the expression \( \frac{q^2 r^4}{q^7 r^3} \), you subtract the exponents of q to get q^

.

Multiple Select

Which of the following properties are used in the simplification of \( \left( \frac{-2a^4}{b^2} \right)^3 \)?

• Power of a quotient

• Power of a product

• Power of a sum

• Power of a power

Open Ended

Determine whether the following expression is a polynomial and, if so, state its degree: \( \frac{1}{4}x^4y^3 - 8x^5 \).

Multiple Choice

What is the simplified form of the expression \((4x^2 - 5x + 6) - (2x^2 + 3x - 1)\)?

• 2x^2 - 8x + 7

• 2x^2 - 2x + 5

• 6x^2 - 2x + 5

• 2x^2 + 8x - 7

Multiple Choice

Which of the following expressions correctly represents the result of multiplying 3x by (2x^2 - 4x + 6)?

• 6x^3 - 12x^2 + 18x

• 6x^2 - 12x + 18

• 3x^2 - 4x + 6

• 6x^3 - 12x + 18x

Open Ended

Explain how you can determine whether an expression is a polynomial and how to find its degree. Use examples to support your explanation.

Fill in the Blanks

The polynomial that represents the distance driven in the real-world example is

- 10x.

Multiple Select

Which of the following steps are necessary when multiplying two polynomials such as (n^2 + 4n - 6)(n + 2)?

• Apply the distributive property

• Multiply each term in the first polynomial by each term in the second

• Combine like terms

• Divide each term by the highest degree

Open Ended

How do the properties of exponents help simplify expressions involving monomials?

Multiple Choice

Which property of exponents allows you to rewrite (x^a)^b as x^(ab)?

• Product of Powers

• Power of a Power

• Negative Exponent

• Zero Power