Intermediate Algebra: Polynomial Operations, Synthetic Division, Remainder Theorem, and Equations

Q11. Simplify: 7x^7y^2 \div 6y^2

Background

Topic: Simplifying Rational Expressions

This question tests your ability to divide monomials and simplify expressions by canceling common factors.

Key Terms and Formulas:

• Monomial: An algebraic expression with one term.

• To divide monomials: Divide coefficients and subtract exponents of like variables.

Step-by-Step Guidance

1. Write the expression as a fraction: $\frac{7x^7y^2}{6y^2}$

2. Divide the coefficients: $\frac{7}{6}$

3. Subtract the exponents of $y$: $y^{2-2} = y^0$

4. Remember that $y^0 = 1$, so it can be omitted from the final expression.

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Q12. Divide: (-25x^3 - 30x^2 - 13x + 7) \div (5x + 4)

Background

Topic: Polynomial Long Division

This question is about dividing a cubic polynomial by a linear binomial using long division.

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. Set up the division: $\frac{-25x^3 - 30x^2 - 13x + 7}{5x + 4}$

2. Divide the leading term of the dividend by the leading term of the divisor: $\frac{-25x^3}{5x}$

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

4. Repeat the process with the new polynomial (the result after subtraction) until the degree of the remainder is less than the degree of the divisor.

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Q13. Add: (12x^3 + x^2 - 84x - 7) + (6x^2 - 42)

Background

Topic: Adding Polynomials

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

Key Terms and Formulas:

• Like terms: Terms with the same variable and exponent.

• To add polynomials: Add coefficients of like terms.

Step-by-Step Guidance

1. Write both polynomials in standard form and align like terms:

2. Combine the $x^3$ terms, $x^2$ terms, $x$ terms, and constants.

3. Add the coefficients for each group of like terms.

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Q14. Use synthetic division to divide: 18x^9 \div x

Background

Topic: Synthetic Division

This question asks you to use synthetic division, which is typically used for dividing polynomials by linear factors of the form $(x - c)$.

Key Terms and Formulas:

• Synthetic division: A shortcut for dividing a polynomial by a linear factor.

• For $x$ as the divisor, $c = 0$ in $x - c$.

Step-by-Step Guidance

1. Set up synthetic division for $c = 0$ (since dividing by $x$ is the same as $x - 0$).

2. Write the coefficients of $18x^9$ (which are 18 followed by eight zeros for missing terms).

3. Carry down the leading coefficient and perform synthetic division steps.

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Q15. For $P(x) = x^2 + 13x + 31$, use the remainder theorem to find $P(-9)$

Background

Topic: Remainder Theorem

This question tests your understanding of the remainder theorem, which states that the remainder of dividing $P(x)$ by $x - c$ is $P(c)$.

Key Terms and Formulas:

• Remainder Theorem: $P(c)$ is the remainder when $P(x)$ is divided by $x - c$.

Step-by-Step Guidance

1. Identify $c = -9$.

2. Substitute $x = -9$ into $P(x)$: $P(-9) = (-9)^2 + 13(-9) + 31$

3. Calculate each term separately: $(-9)^2$, $13 \times (-9)$, and $31$.

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Q16. Solve the equation: $x^2 = 9$

Background

Topic: Solving Quadratic Equations

This question tests your ability to solve a basic quadratic equation by taking square roots.

Key Terms and Formulas:

• Quadratic equation: An equation of the form $ax^2 + bx + c = 0$.

• Square root property: If $x^2 = a$, then $x = \pm\sqrt{a}$.

Step-by-Step Guidance

1. Recognize that $x^2 = 9$ can be solved by taking the square root of both sides.

2. Apply the square root property: $x = \pm\sqrt{9}$

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Q17. Solve: $x^2 + 9x + 20 = 0$

Background

Topic: Solving Quadratic Equations by Factoring

This question tests your ability to factor a quadratic equation and solve for $x$.

Key Terms and Formulas:

• Factoring: Writing a quadratic as $(x + a)(x + b) = 0$.

• Zero Product Property: If $ab = 0$, then $a = 0$ or $b = 0$.

Step-by-Step Guidance

1. Write the equation in standard form: $x^2 + 9x + 20 = 0$

2. Look for two numbers that multiply to $20$and add to $9$.

3. Factor the quadratic into two binomials.

4. Set each factor equal to zero and solve for $x$.

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Q19. A recent advertisement claims that 2 out of every 7 doctors recommend a supplement to increase energy levels. If a local hospital employs 119 doctors, how many would you expect to recommend the supplement? (Round your answer to the nearest whole number, if necessary.)

Background

Topic: Proportions and Applications

This question tests your ability to set up and solve a proportion based on a real-world scenario.

Key Terms and Formulas:

• Proportion: An equation stating that two ratios are equal.

• Set up the proportion: $\frac{2}{7} = \frac{x}{119}$

Step-by-Step Guidance

1. Let $x$ be the number of doctors at the hospital who recommend the supplement.

2. Set up the proportion: $\frac{2}{7} = \frac{x}{119}$

3. Cross-multiply to solve for $x$: $2 \times 119 = 7 \times x$

4. Isolate $x$ by dividing both sides by $7$.

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