Math Problem Statement

-12t^7 18t^9 over - 6t^3 simplified is? -3t^12 - 12t^7, - 2t^3 - 3t, - 3t^6(4t-1), t^4(2-3t^2). Which answer is correct

Solution

Let's simplify the given expression step by step:

The expression is:

12t7+18t96t3\frac{-12t^7 + 18t^9}{-6t^3}

Step 1: Simplify each term by dividing by 6t3-6t^3.

For the first term, 12t76t3\frac{-12t^7}{-6t^3}:

126=2andt7t3=t73=t4\frac{-12}{-6} = 2 \quad \text{and} \quad \frac{t^7}{t^3} = t^{7-3} = t^4

So, 12t76t3=2t4\frac{-12t^7}{-6t^3} = 2t^4.

For the second term, 18t96t3\frac{18t^9}{-6t^3}:

186=3andt9t3=t93=t6\frac{18}{-6} = -3 \quad \text{and} \quad \frac{t^9}{t^3} = t^{9-3} = t^6

So, 18t96t3=3t6\frac{18t^9}{-6t^3} = -3t^6.

Step 2: Combine the results.

2t43t62t^4 - 3t^6

Now let's check the answer options:

  1. 3t1212t7-3t^{12} - 12t^7 - This is not correct.
  2. 2t33t-2t^3 - 3t - This is not correct.
  3. 3t6(4t1)-3t^6(4t - 1) - Let's expand this:

3t6(4t1)=12t7+3t6-3t^6(4t - 1) = -12t^7 + 3t^6

This does not match 2t43t62t^4 - 3t^6.

  1. t4(23t2)t^4(2 - 3t^2) - Let's expand this:

t4(23t2)=2t43t6t^4(2 - 3t^2) = 2t^4 - 3t^6

This matches the simplified expression.

Correct Answer:

The correct answer is:

t4(23t2)\boxed{t^4(2 - 3t^2)}

Would you like further details on any of these steps?

Related Questions:

  1. How do you factor higher-degree polynomials like t4(23t2)t^4(2 - 3t^2)?
  2. What are the rules for simplifying exponents during division?
  3. How do you apply the distributive property when expanding terms?
  4. Can you explain more about combining like terms in polynomial expressions?
  5. What are some common mistakes to avoid when simplifying expressions with negative signs?

Tip:

Always watch out for signs and double-check your exponent rules when simplifying expressions involving variables.

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Math Problem Analysis

Mathematical Concepts

Algebra
Simplifying Rational Expressions
Polynomial Division
Exponent Rules

Formulas

Division of powers with the same base: a^m / a^n = a^(m-n)
Distributive property: a(b + c) = ab + ac

Theorems

Exponentiation Laws
Distributive Property

Suitable Grade Level

Grades 9-12