Examkin

Question

Difficulty: MediumSimplifying Expressions and Combining Like Terms

Match each of the unsimplified algebraic expressions on the left with its equivalent simplified form on the right. (Assume all variables represent real numbers.)

  • 3r(2r−s)−2s(r−3s)3r(2r - s) - 2s(r - 3s)6r2−5rs+6s26r^2 - 5rs + 6s^2
  • (2r−s)2−2(r2−rs)(2r - s)^2 - 2(r^2 - rs)2r2−2rs+s22r^2 - 2rs + s^2
  • 12(4r2−6rs)−(r−s)(2r+3s)\frac{1}{2}(4r^2 - 6rs) - (r - s)(2r + 3s)3s2−4rs3s^2 - 4rs
  • r2(6−s)−s(r2−5s)r^2(6 - s) - s(r^2 - 5s)6r2−2r2s+5s26r^2 - 2r^2s + 5s^2

Answer

The correct matches pair the first expression with 6r2−5rs+6s26r^2 - 5rs + 6s^2, the second expression with 2r2−2rs+s22r^2 - 2rs + s^2, the third expression with 3s2−4rs3s^2 - 4rs, and the fourth expression with 6r2−2r2s+5s26r^2 - 2r^2s + 5s^2.
Each expression is correctly simplified by performing distribution first (carefully tracking negative signs and binomial expansion rules) and then combining terms that share the exact same variable powers.

Step-by-Step Solution

1
Simplify the expression 3r(2r−s)−2s(r−3s)3r(2r - s) - 2s(r - 3s).
6r2−5rs+6s26r^2 - 5rs + 6s^2
Distribute 3r3r to get 6r2−3rs6r^2 - 3rs. Then distribute −2s-2s to get −2rs+6s2-2rs + 6s^2 (noting that a negative times a negative is positive). Combine the like terms −3rs-3rs and −2rs-2rs to get −5rs-5rs.
2
Simplify the expression (2r−s)2−2(r2−rs)(2r - s)^2 - 2(r^2 - rs).
2r2−2rs+s22r^2 - 2rs + s^2
Square the binomial (2r−s)2(2r - s)^2 using the formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2 to get 4r2−4rs+s24r^2 - 4rs + s^2. Distribute −2-2 to get −2r2+2rs-2r^2 + 2rs. Combine 4r2−2r24r^2 - 2r^2 to get 2r22r^2, and −4rs+2rs-4rs + 2rs to get −2rs-2rs.
3
Simplify the expression 12(4r2−6rs)−(r−s)(2r+3s)\frac{1}{2}(4r^2 - 6rs) - (r - s)(2r + 3s).
3s2−4rs3s^2 - 4rs
Distribute 12\frac{1}{2} to get 2r2−3rs2r^2 - 3rs. Expand the binomial product to get 2r2+3rs−2rs−3s2=2r2+rs−3s22r^2 + 3rs - 2rs - 3s^2 = 2r^2 + rs - 3s^2. Subtract this entire expression: (2r2−3rs)−(2r2+rs−3s2)=2r2−3rs−2r2−rs+3s2(2r^2 - 3rs) - (2r^2 + rs - 3s^2) = 2r^2 - 3rs - 2r^2 - rs + 3s^2. Combine like terms to get −4rs+3s2-4rs + 3s^2.
4
Simplify the expression r2(6−s)−s(r2−5s)r^2(6 - s) - s(r^2 - 5s).
6r2−2r2s+5s26r^2 - 2r^2s + 5s^2
Distribute r2r^2 to get 6r2−r2s6r^2 - r^2s. Distribute −s-s to get −sr2+5s2-sr^2 + 5s^2. Combine the like terms −r2s-r^2s and −sr2-sr^2 (since multiplication is commutative) to get −2r2s-2r^2s.

Key Concept

Simplifying expressions by applying the distributive property, expanding products of binomials, and combining like terms.
Rate this question

Topics

Simplifying Expressions and Combining Like Terms

More questions from this topic