Multiplying Polynomials Worksheets - Free Printable

The image you provided shows a worksheet with several algebraic expressions involving monomials. The task appears to be simplifying or performing operations (such as addition, subtraction, multiplication, or division) on these monomials. Below, I will solve each problem step by step and explain the solution.

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Problem 1:

$$
\frac{2}{3}x^4 - \frac{1}{6}x^4
$$

#### Solution:
1. Both terms have the same variable part \( x^4 \), so we can combine them by subtracting their coefficients.
2. The coefficients are \( \frac{2}{3} \) and \( -\frac{1}{6} \). To subtract these fractions, we need a common denominator. The least common denominator of 3 and 6 is 6.
3. Rewrite the fractions with the common denominator:
$$
\frac{2}{3} = \frac{2 \cdot 2}{3 \cdot 2} = \frac{4}{6}
$$
So the expression becomes:
$$
\frac{4}{6}x^4 - \frac{1}{6}x^4
$$
4. Subtract the numerators:
$$
\frac{4}{6} - \frac{1}{6} = \frac{4 - 1}{6} = \frac{3}{6}
$$
5. Simplify the fraction:
$$
\frac{3}{6} = \frac{1}{2}
$$
6. Therefore, the simplified expression is:
$$
\frac{1}{2}x^4
$$

#### Final Answer for Problem 1:
$$
\boxed{\frac{1}{2}x^4}
$$

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Problem 2:

$$
\frac{2}{3}x^4 + \frac{1}{6}x^4
$$

#### Solution:
1. Both terms have the same variable part \( x^4 \), so we can combine them by adding their coefficients.
2. The coefficients are \( \frac{2}{3} \) and \( \frac{1}{6} \). To add these fractions, we need a common denominator. The least common denominator of 3 and 6 is 6.
3. Rewrite the fractions with the common denominator:
$$
\frac{2}{3} = \frac{2 \cdot 2}{3 \cdot 2} = \frac{4}{6}
$$
So the expression becomes:
$$
\frac{4}{6}x^4 + \frac{1}{6}x^4
$$
4. Add the numerators:
$$
\frac{4}{6} + \frac{1}{6} = \frac{4 + 1}{6} = \frac{5}{6}
$$
5. Therefore, the simplified expression is:
$$
\frac{5}{6}x^4
$$

#### Final Answer for Problem 2:
$$
\boxed{\frac{5}{6}x^4}
$$

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Problem 3:

$$
-\frac{2}{3}x^4 - \frac{1}{6}x^4
$$

#### Solution:
1. Both terms have the same variable part \( x^4 \), so we can combine them by subtracting their coefficients.
2. The coefficients are \( -\frac{2}{3} \) and \( -\frac{1}{6} \). To subtract these fractions, we need a common denominator. The least common denominator of 3 and 6 is 6.
3. Rewrite the fractions with the common denominator:
$$
-\frac{2}{3} = -\frac{2 \cdot 2}{3 \cdot 2} = -\frac{4}{6}
$$
So the expression becomes:
$$
-\frac{4}{6}x^4 - \frac{1}{6}x^4
$$
4. Subtract the numerators:
$$
-\frac{4}{6} - \frac{1}{6} = \frac{-4 - 1}{6} = \frac{-5}{6}
$$
5. Therefore, the simplified expression is:
$$
-\frac{5}{6}x^4
$$

#### Final Answer for Problem 3:
$$
\boxed{-\frac{5}{6}x^4}
$$

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Problem 4:

$$
-\frac{2}{3}x^4 + \frac{1}{6}x^4
$$

#### Solution:
1. Both terms have the same variable part \( x^4 \), so we can combine them by adding their coefficients.
2. The coefficients are \( -\frac{2}{3} \) and \( \frac{1}{6} \). To add these fractions, we need a common denominator. The least common denominator of 3 and 6 is 6.
3. Rewrite the fractions with the common denominator:
$$
-\frac{2}{3} = -\frac{2 \cdot 2}{3 \cdot 2} = -\frac{4}{6}
$$
So the expression becomes:
$$
-\frac{4}{6}x^4 + \frac{1}{6}x^4
$$
4. Add the numerators:
$$
-\frac{4}{6} + \frac{1}{6} = \frac{-4 + 1}{6} = \frac{-3}{6}
$$
5. Simplify the fraction:
$$
\frac{-3}{6} = -\frac{1}{2}
$$
6. Therefore, the simplified expression is:
$$
-\frac{1}{2}x^4
$$

#### Final Answer for Problem 4:
$$
\boxed{-\frac{1}{2}x^4}
$$

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Problem 5:

$$
\frac{2}{3}x^4 - \frac{1}{6}x^4
$$

This is the same as Problem 1, so the solution is:
$$
\boxed{\frac{1}{2}x^4}
$$

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Problem 6:

$$
\frac{2}{3}x^4 + \frac{1}{6}x^4
$$

This is the same as Problem 2, so the solution is:
$$
\boxed{\frac{5}{6}x^4}
$$

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Final Answers:

1. \( \boxed{\frac{1}{2}x^4} \)
2. \( \boxed{\frac{5}{6}x^4} \)
3. \( \boxed{-\frac{5}{6}x^4} \)
4. \( \boxed{-\frac{1}{2}x^4} \)
5. \( \boxed{\frac{1}{2}x^4} \)
6. \( \boxed{\frac{5}{6}x^4} \)

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If you have any further questions or need additional clarification, feel free to ask!