Q1a. Simplify: (Write your answer without negative exponents.)

Background

Topic: Laws of Exponents and Simplifying Expressions

This question tests your ability to apply the rules of exponents, including multiplying powers with the same base and handling negative exponents.

Key Terms and Formulas:

• Product of Powers:

• Negative Exponent:

• Any variable to the zero power: (if )

Step-by-Step Guidance

1. Expand the expression by multiplying the coefficients: .

2. Combine like bases using the product of powers rule: , , .

3. Simplify each variable's exponent: add exponents for each base.

4. Rewrite any negative exponents as positive by moving them to the denominator.

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Q1b. Simplify: (Write your answer without negative exponents.)

Background

Topic: Exponent Rules and Simplifying Expressions

This question tests your understanding of exponent rules, including raising a power to a power and multiplying expressions with exponents.

Key Terms and Formulas:

• Power of a Power:

• Product of Powers:

• Negative Exponent:

Step-by-Step Guidance

1. Apply the power to each factor inside the first parentheses: .

2. Multiply this result by .

3. Combine like bases: and , and .

4. Express the final answer with only positive exponents.

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Q2a. Simplify:

Background

Topic: Combining Like Terms (Polynomials)

This question tests your ability to add polynomials by combining like terms.

Key Terms and Formulas:

• Like Terms: Terms with the same variable(s) raised to the same power(s).

• Add/Subtract Like Terms: Combine coefficients of like terms.

Step-by-Step Guidance

1. Write both polynomials in a single expression: .

2. Group like terms: and , and , and .

3. Add the coefficients for each variable.

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Q3a. Divide using long division:

Background

Topic: Polynomial Long Division

This question tests your ability to divide polynomials using the long division algorithm.

Key Terms and Formulas:

• Dividend: The polynomial being divided ().

• Divisor: The polynomial you are dividing by ().

• Quotient: The result of the division.

• Remainder: What is left after division.

Step-by-Step Guidance

1. Arrange the dividend in descending order of powers: .

2. Divide the leading term of the dividend by the leading term of the divisor: .

3. Multiply the entire divisor by the result from step 2 and subtract from the dividend.

4. Repeat the process with the new polynomial until the degree of the remainder is less than the degree of the divisor.

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Q4a. Divide using synthetic division:

Background

Topic: Synthetic Division

This question tests your ability to use synthetic division to divide a polynomial by a linear factor.

Key Terms and Formulas:

• Synthetic Division: A shortcut for dividing a polynomial by a binomial of the form .

• Arrange the polynomial in standard form and fill in any missing terms with zero coefficients.

Step-by-Step Guidance

1. Rewrite the polynomial in standard form: .

2. Set to find for synthetic division.

3. List the coefficients: .

4. Carry out synthetic division using and the coefficients.

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Q5. Use the Remainder Theorem to determine whether is a factor of

Background

Topic: Remainder Theorem and Factor Theorem

This question tests your ability to use the Remainder Theorem to check if a binomial is a factor of a polynomial.

Key Terms and Formulas:

• Remainder Theorem: The remainder when is divided by is .

• If , then is a factor of .

Step-by-Step Guidance

1. Set to find .

2. Substitute into .

3. Calculate to determine if the remainder is zero.

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Q6a. Factor completely:

Background

Topic: Factoring Polynomials

This question tests your ability to factor polynomials by grouping and other factoring techniques.

Key Terms and Formulas:

• Factoring by Grouping: Group terms to factor common factors, then factor further if possible.

• Check your factorization by multiplying (FOIL or distribution).

Step-by-Step Guidance

1. Group the polynomial into two pairs: and .

2. Factor out the greatest common factor (GCF) from each group.

3. Look for a common binomial factor and factor it out.

4. Check if the resulting factors can be factored further.

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Q7a. Solve by factoring:

Background

Topic: Solving Quadratic Equations by Factoring

This question tests your ability to solve quadratic equations by factoring and using the zero product property.

Key Terms and Formulas:

• Zero Product Property: If , then or .

• Factoring: Express the equation as a product of factors set equal to zero.

Step-by-Step Guidance

1. Move all terms to one side to set the equation to zero: .

2. Factor out the greatest common factor from the equation.

3. Set each factor equal to zero and solve for .

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Q8a. Solve using the quadratic formula:

Background

Topic: Quadratic Formula

This question tests your ability to solve quadratic equations using the quadratic formula.

Key Terms and Formulas:

• Quadratic Formula:

• Standard Form:

Step-by-Step Guidance

1. Identify , , and from the equation: .

2. Write the quadratic formula and substitute the values for , , and .

3. Calculate the discriminant: .

4. Set up the expression for using the quadratic formula, but do not simplify to the final values yet.

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