Question

State whether each sentence is true or false. If false, replace the underlined word, phrase, expression, or number to make a true sentence.

The least common multiply for$x^2-25$ and$x-5$ isrole="math" localid="1648029600853" $\underset{¯}{x-5}$.

Short answer

The given statement is false.

The corrected statement is “The least common multiply for$x^2-25$ and $x-5$is$\underset{¯}{x^2-25}$ .”.

Step-by-step solution

Step 1. State the concept of least common multiple.

The least common multiple (LCM) is the least number that is a multiple of two or more numbers or polynomials.

Step 2. Explanation of the statement.

The expression is as follows $x^2-25$ and$x-5$.

Calculate the prime factors of both the expressions:

role="math" localid="1648029852221" $$\begin{gathered} x^2-25=\left(x+5\right)\cdot \left(x-5\right) \\ x-5=1.\left(x-5\right) \end{gathered}$$

Use each prime factor $\left(x+5\right)$and $\left(x-5\right)$, the greatest number of times it appears in either of the factorizations.

role="math" localid="1648029943926" $$\begin{gathered} \text{LCM}=\left(x+5\right)\cdot \left(x-5\right) \\ =x^2-25 \end{gathered}$$

Step 3. Write whether the statement is true or false.

The given statement “The least common multiply for$x^2-25$and $x-5$is .localid="1648030000920" $\underset{¯}{x-5}$” is false and the corrected statement will be “The least common multiply for$x^2-25$and $x-5$is $\underset{¯}{x^2-25}$.”.