Q1. (x^2 - 3x + 2) - (x - 4x^2)

Background

Topic: Polynomial Operations

This question tests your ability to subtract polynomials and combine like terms.

Key Terms and Formulas

• Polynomial: An expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication.

• Like terms: Terms with the same variable raised to the same power.

Step-by-Step Guidance

1. Write both polynomials in standard form (descending powers of x).

2. Distribute the negative sign to each term in the second polynomial.

3. Combine like terms by adding or subtracting the coefficients of terms with the same degree.

4. Write the resulting polynomial in standard form.

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Q2. (6r - 5)^2

Background

Topic: Expanding Binomials

This question tests your ability to expand the square of a binomial using the distributive property or the binomial formula.

Key Terms and Formulas

• Binomial Square Formula:

Step-by-Step Guidance

1. Identify and in the expression .

2. Apply the binomial square formula: .

3. Calculate each term: , , and .

4. Combine the terms to write the expanded expression.

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Q3. (t + 2)(3t^2 - t + 4)

Background

Topic: Multiplying Polynomials

This question tests your ability to multiply a binomial by a trinomial.

Key Terms and Formulas

• Distributive Property:

Step-by-Step Guidance

1. Distribute each term in the binomial to each term in the trinomial .

2. Multiply by each term in the trinomial.

3. Multiply $2$ by each term in the trinomial.

4. Add the results together and combine like terms.

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Q4. \( \frac{2x^3 - 11x^2 + 28}{x - 5} \)

Background

Topic: Polynomial Division

This question tests your ability to divide a cubic polynomial by a linear binomial using either long division or synthetic division.

Key Terms and Formulas

• Polynomial Division: Dividing one polynomial by another, often using long division or synthetic division.

Step-by-Step Guidance

1. Set up the division, identifying the divisor and the dividend .

2. If using synthetic division, use as the synthetic divisor.

3. Write the coefficients of the dividend in order, including zeros for missing terms.

4. Perform the synthetic division steps: bring down the first coefficient, multiply, add, and repeat for each column.

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Q5. Use synthetic division to perform: \( \frac{3x^3 + 4x^2 - 9x + 6}{x + 2} \)

Background

Topic: Synthetic Division

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

Key Terms and Formulas

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

Step-by-Step Guidance

1. Set up synthetic division using (since the divisor is ).

2. List the coefficients: .

3. Carry out the synthetic division steps: bring down the first coefficient, multiply, add, and repeat.

4. Interpret the result as the coefficients of the quotient and the remainder.

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Q6. Factor completely: 6x^2 - 17x + 7

Background

Topic: Factoring Quadratic Polynomials

This question tests your ability to factor a quadratic trinomial into the product of two binomials.

Key Terms and Formulas

• Factoring: Writing a polynomial as a product of its factors.

• Quadratic trinomial: A polynomial of the form .

Step-by-Step Guidance

1. Identify , , .

2. Look for two numbers that multiply to and add to .

3. Rewrite the middle term using these two numbers and factor by grouping.

4. Factor out the greatest common factor from each group and write the final factored form.

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Q7. Solve: |x + 4| = 7

Background

Topic: Absolute Value Equations

This question tests your ability to solve equations involving absolute values.

Key Terms and Formulas

• Absolute Value: is the distance of from zero on the number line.

• Solving : or

Step-by-Step Guidance

1. Set up two equations: and .

2. Solve each equation for .

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Q8. Graph the quadratic function . Give the intercepts, vertex, axis, domain, range, and the largest open intervals of the domain over which the function is increasing or decreasing.

Background

Topic: Quadratic Functions and Graphing

This question tests your understanding of the properties of quadratic functions, including graphing, finding intercepts, vertex, axis of symmetry, domain, range, and intervals of increase/decrease.

Key Terms and Formulas

• Vertex form:

• Vertex:

• Axis of symmetry:

• Domain: All real numbers for quadratic functions.

• Range: Depends on the direction the parabola opens.

Step-by-Step Guidance

1. Identify , , and in the quadratic function.

2. Find the vertex using and substitute back to find .

3. Find the y-intercept by evaluating .

4. Find the x-intercepts by solving .

5. Determine the axis of symmetry, domain, and range.

6. Describe the intervals where the function is increasing or decreasing.

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