Polynomial Factoring: True Or False?

Let's dive into some polynomial expressions and determine whether the given factorizations are accurate. We'll go through each statement, carefully applying the distributive property to check if the factored form matches the original polynomial. Understanding polynomial factorization is a fundamental skill in algebra, with applications ranging from solving equations to simplifying complex expressions. So, let's get started and test your knowledge!

Analyzing the Statements

Statement 1: $15 m^3 - 6m = 3m(5m^2 - 6m)$

To verify if this statement is true, we need to distribute the $3m$ across the terms inside the parenthesis on the right side of the equation. This means we multiply $3m$ by both $5m^2$ and $-6m$. Let's perform the multiplication:

$3m * 5m^2 = 15m^3$

$3m * -6m = -18m^2$

Combining these results, we get:

$3m(5m^2 - 6m) = 15m^3 - 18m^2$

Now, let's compare this result with the left side of the original equation, which is $15m^3 - 6m$. We can see that the two sides are not equal because $15m^3 - 18m^2$ is not the same as $15m^3 - 6m$. The term $-18m^2$ is different from $-6m$.

Therefore, the first statement, $15 m^3 - 6m = 3m(5m^2 - 6m)$, is false.

Key Takeaway: When factoring, always double-check your work by distributing the terms back into the parenthesis to ensure the result matches the original expression. A simple mistake in multiplication can lead to an incorrect factorization. In this case, the incorrect multiplication of $3m$ with $-6m$ led to $-18m^2$ instead of $-6m$.

Statement 2: $32 m^4 + 12 m^3 = 4 m^3(8 m + 3)$

To check the validity of this statement, we will again distribute the term outside the parenthesis, which is $4m^3$, across the terms inside the parenthesis, $(8m + 3)$. Let's do the multiplication:

$4m^3 * 8m = 32m^4$

$4m^3 * 3 = 12m^3$

Combining these, we have:

$4m^3(8m + 3) = 32m^4 + 12m^3$

Now, let's compare this result with the left side of the original equation, which is $32m^4 + 12m^3$. We can observe that the two sides are indeed equal:

$32m^4 + 12m^3 = 32m^4 + 12m^3$

Therefore, the second statement, $32 m^4 + 12 m^3 = 4 m^3(8 m + 3)$, is true.

Key Takeaway: Factoring involves identifying common factors and extracting them from the terms of a polynomial. To verify the factoring, distribute the extracted factors back into the parenthesis and ensure that the result matches the original polynomial. If the distribution yields the original polynomial, the factoring is correct.

Statement 3: $40 m^6 - 4 = 4(10 m^6 - 1)$

To determine if this statement is true, we distribute the $4$ on the right side of the equation across the terms inside the parenthesis, $(10m^6 - 1)$. Performing the multiplication:

$4 * 10m^6 = 40m^6$

$4 * -1 = -4$

Combining these, we get:

$4(10m^6 - 1) = 40m^6 - 4$

Now, let's compare this result with the left side of the original equation, which is $40m^6 - 4$. We can see that the two sides are equal:

$40m^6 - 4 = 40m^6 - 4$

Therefore, the third statement, $40 m^6 - 4 = 4(10 m^6 - 1)$, is true.

Key Takeaway: Factoring out a constant term is a straightforward process. Simply divide each term in the original polynomial by the constant factor and place the constant outside the parenthesis. Always verify your work by distributing the constant back into the parenthesis to ensure that the result matches the original polynomial.

Statement 4: $6 m^2 + 18 m = 6 m^2(1 + 3 m)$

To check the validity of this statement, we distribute the term outside the parenthesis, which is $6m^2$, across the terms inside the parenthesis, $(1 + 3m)$. Let's do the multiplication:

$6m^2 * 1 = 6m^2$

$6m^2 * 3m = 18m^3$

Combining these, we have:

$6m^2(1 + 3m) = 6m^2 + 18m^3$

Now, let's compare this result with the left side of the original equation, which is $6m^2 + 18m$. We can observe that the two sides are not equal because $6m^2 + 18m^3$ is not the same as $6m^2 + 18m$. The term $18m^3$ is different from $18m$.

Therefore, the fourth statement, $6 m^2 + 18 m = 6 m^2(1 + 3 m)$, is false.

Key Takeaway: When factoring out a term with a variable, it's essential to correctly account for the exponents. Distributing the factored term back into the parenthesis helps identify any mistakes. In this case, the incorrect multiplication of $6m^2$ with $3m$ led to $18m^3$ instead of $18m$.

Conclusion

In summary, after carefully analyzing each statement, we find that:

• Statement 1 is false. $15 m^3 - 6m ≠ 3m(5m^2 - 6m)$
• Statement 2 is true. $32 m^4 + 12 m^3 = 4 m^3(8 m + 3)$
• Statement 3 is true. $40 m^6 - 4 = 4(10 m^6 - 1)$
• Statement 4 is false. $6 m^2 + 18 m ≠ 6 m^2(1 + 3 m)$

Remember, the key to verifying polynomial factorizations is to distribute the factored terms back into the parenthesis and compare the result with the original polynomial. Practice makes perfect, so keep honing your skills in factoring and simplifying algebraic expressions.

For further information on polynomial factorization, you can visit Khan Academy's Polynomial Factorization Section.