Question

Challenge Problems. $$p^{3} q^{3}+p q^{2} r^{3}-p^{2} r^{4} \mathrm{by}\left(-p^{2} r\right)$$

Short answer

\(-p^5q^3r - p^3q^2r^4 + p^4r^5\)

Step-by-step solution

Apply the distributive property

To distribute \( -p^2r \) across \( p^3q^3 + pq^2r^3 - p^2r^4 \) we multiply \( -p^2r \) by each term inside the parentheses separately.

Multiply the first term

Multiply \( -p^2r \) by \( p^3q^3 \) using the properties of exponentiation: \( (-p^2r)(p^3q^3) = -p^{2+3}r^{1+0}q^3 = -p^5q^3r \).

Multiply the second term

Multiply \( -p^2r \) by \( pq^2r^3 \) using the properties of exponentiation: \( (-p^2r)(pq^2r^3) = -p^{2+1}r^{1+3}q^2 = -p^3q^2r^4 \).

Multiply the third term

Multiply \( -p^2r \) by \( -p^2r^4 \) using the properties of exponentiation: \( (-p^2r)(-p^2r^4) = p^{2+2}r^{1+4} = p^4r^5 \).

Combine the results

Combine all the products from the previous steps to get the final result: \( -p^5q^3r - p^3q^2r^4 + p^4r^5 \).

Key concepts

Properties of Exponentiation

Understanding the properties of exponentiation is crucial when dealing with polynomials involving powers. Let's clarify these properties with an easy-to-grasp explanation. When multiplying two algebraic terms with the same base, we simply add their exponents. For example, if we multiply p^2 and p^3, we get p^(2+3) or p^5. It's essential to recognize that this addition of exponents applies only when the bases are the same and we are multiplying, not adding, the terms.

When it comes to dividing terms with exponents, the process is just the opposite. We subtract the exponent of the denominator from the exponent of the numerator as long as we have the same base. This set of rules helps ensure consistency and correctness when handling more complex algebraic expressions with exponents.

Polynomial Multiplication

Multiplying Monomials and Polynomials

Polynomial multiplication might seem daunting, but it's just an extension of simpler multiplication we use every day. To multiply a monomial, a single term, by a polynomial, several terms added or subtracted together, we use the distributive property. This property states that you distribute the multiplication of the single term over all terms in the polynomial. Imagine we're giving out cookies to a group of friends; we have to give the same number of cookies to each person to be fair—that's essentially what we're doing with terms in the polynomial.

Example: When we multiply (-p^2r) by p^3q^3, the distributive property tells us to multiply the p^2 part and the r part by separate terms in the polynomial, which simplifies the multiplication process and prevents errors.

Algebraic Expressions

Algebraic expressions are like sentences in the language of mathematics, where numbers and variables mix to convey a certain meaning. They can be as simple as x + 5 or as intricate as a polynomial. The key in algebraic expressions is to understand that variables represent unknown quantities that we're trying to figure out or use in calculations. An expression may contain constants, coefficients, variables, operators (such as + or -), and exponents.

When we deal with algebraic expressions, we perform operations according to the established algebraic rules. For instance, we always deal with multiplication and division before we tackle addition and subtraction unless parentheses rearrange the order of operations. Recognizing and correctly handling these various components of algebraic expressions ensure accurate solutions to algebraic problems.

Combining Like Terms

What Are Like Terms?

Like terms are the terms within an algebraic expression that have the same variables raised to the same powers, even though their coefficients might differ. For instance, 3a^2b and -5a^2b are like terms because they have the identical variable part, a^2b.

Why Combine Like Terms?

Combining like terms is a way to simplify an algebraic expression, making it easier to understand and solve. It's like organizing a room: putting books with books and clothes with clothes makes it simpler to see what you have and find what you need. In algebra, combining like terms makes your math 'room' tidier. When we add or subtract like terms, we just add or subtract the coefficients, leaving the variable part unchanged. This is a foundational skill for algebra students, as it helps to simplify expressions and solve equations.