Q1. Divide: (12x^5 + 4x^4 - 12x^2 - 4x) ÷ (3x + 1)

Background

Topic: Polynomial Long Division

This question tests your ability to divide a polynomial by a binomial using long division. This is a fundamental skill in Intermediate Algebra, especially when simplifying expressions or solving equations.

Key Terms and Formulas

• Dividend: The polynomial being divided,

• Divisor: The binomial

• Quotient: The result of the division

• Remainder: Any leftover part after division

Step-by-Step Guidance

1. Arrange the dividend and divisor in standard form, making sure all powers of are present (add zero coefficients if needed).

2. Divide the leading term of the dividend () by the leading term of the divisor () to get the first term of the quotient.

3. Multiply the entire divisor by this first term and subtract the result from the dividend.

4. Bring down the next term from the dividend and repeat the process: divide, multiply, subtract.

5. Continue this process until all terms have been brought down and divided. Stop before calculating the final remainder.

Try solving on your own before revealing the answer!

Q2. If a polynomial is divided by x - 6, the quotient is 2x^2 + 4x - 3 and the remainder is 7. Find the original polynomial.

Background

Topic: Polynomial Division and Remainder Theorem

This question tests your understanding of how to reconstruct the original polynomial from its quotient and remainder after division by a linear binomial.

Key Terms and Formulas

• Quotient:

• Divisor:

• Remainder: $7$

• Original Polynomial:

Step-by-Step Guidance

1. Recall the division algorithm for polynomials: .

2. Substitute the given quotient and remainder into the formula.

3. Expand using the distributive property.

4. Add the remainder $7$ to the expanded expression to get the original polynomial.

5. Stop before combining like terms for the final polynomial expression.

Try solving on your own before revealing the answer!