Step-by-Step Guidance for Multiplying Polynomials (College Algebra)

Q1. Barb sets up the expression (x + 4)(x + 2) to find the area. Barb thinks the product will be x^2 + 8. Explain how Aunt Barbara may have found this expression for the area.

Background

Topic: Multiplying Binomials

This question tests your understanding of how to multiply two binomials and recognize common mistakes in polynomial multiplication.

Key Terms and Formulas

• Binomial: A polynomial with two terms (e.g., x + 4).

• Product of Binomials: Use the distributive property (FOIL method).

$(x + 4)(x + 2)$

$= x(x + 2) + 4(x + 2)$

$= x^2 + 2x + 4x + 8$

Step-by-Step Guidance

1. Recall the distributive property: Each term in the first binomial must be multiplied by each term in the second binomial.

2. Multiply x by both x and 2 in the second binomial.

3. Multiply 4 by both x and 2 in the second binomial.

4. Combine like terms to simplify the expression.

Try solving on your own before revealing the answer!

Q2. Use algebra tiles to model the area of the blanket at the right. Record the area as a simplified expression.

Background

Topic: Visualizing Polynomial Multiplication

This question asks you to use a visual model (algebra tiles) to represent the multiplication of two binomials.

Key Terms and Formulas

• Algebra Tiles: Visual representations of terms in a polynomial (e.g., x-tile, unit tile).

• Area Model: Each region represents a term in the expanded product.

$(x + 4)(x + 2)$

Step-by-Step Guidance

1. Draw a rectangle and label one side as x + 4 and the other as x + 2.

2. Divide the rectangle into four regions: x*x, x*2, 4*x, and 4*2.

3. Write the expression for each region and sum them.

4. Combine like terms to simplify the area expression.

Try solving on your own before revealing the answer!

Q3. Did Aunt Barbara correctly find the area of the blanket? Can her method be used when multiplying polynomials?

Background

Topic: Error Analysis in Polynomial Multiplication

This question asks you to evaluate the correctness of a multiplication method and whether it is generally applicable.

Key Terms and Formulas

• Correct Method: Distributive property or FOIL for binomials.

• Common Mistake: Only multiplying the first terms and the constants, missing the cross terms.

Step-by-Step Guidance

1. Review Aunt Barbara's result: x^2 + 8.

2. Compare this to the expanded form using the distributive property.

3. Identify which terms are missing in her calculation.

4. Discuss whether her method works for all polynomial multiplication.

Try solving on your own before revealing the answer!

Q4. (x – 5)(x + 3)

Background

Topic: Multiplying Binomials

This question tests your ability to multiply two binomials using the distributive property.

Key Terms and Formulas

• Distributive Property: Multiply each term in the first binomial by each term in the second binomial.

$(x - 5)(x + 3)$

Step-by-Step Guidance

1. Multiply x by x and by 3.

2. Multiply -5 by x and by 3.

3. Write out all four terms.

4. Combine like terms to simplify.

Try solving on your own before revealing the answer!

Q5. (2x + 7)(3x – 1)

Background

Topic: Multiplying Binomials

This question tests your ability to multiply two binomials with coefficients.

Key Terms and Formulas

• Distributive Property: Multiply each term in the first binomial by each term in the second binomial.

$(2x + 7)(3x - 1)$

Step-by-Step Guidance

1. Multiply 2x by 3x and by -1.

2. Multiply 7 by 3x and by -1.

3. Write out all four terms.

4. Combine like terms to simplify.

Try solving on your own before revealing the answer!

Q6. (3r^2 – 2)(r – 9)

Background

Topic: Multiplying a Binomial by a Binomial (with higher degree)

This question tests your ability to multiply a binomial with a quadratic term by another binomial.

Key Terms and Formulas

• Distributive Property: Each term in the first binomial is multiplied by each term in the second binomial.

$(3r^2 - 2)(r - 9)$

Step-by-Step Guidance

1. Multiply 3r^2 by r and by -9.

2. Multiply -2 by r and by -9.

3. Write out all four terms.

4. Combine like terms to simplify.

Try solving on your own before revealing the answer!

Q7. (-x + 11)(-5x + 8)

Background

Topic: Multiplying Binomials with Negative Coefficients

This question tests your ability to multiply binomials that include negative coefficients.

Key Terms and Formulas

• Distributive Property: Multiply each term in the first binomial by each term in the second binomial.

$(-x + 11)(-5x + 8)$

Step-by-Step Guidance

1. Multiply -x by -5x and by 8.

2. Multiply 11 by -5x and by 8.

3. Write out all four terms.

4. Combine like terms to simplify.

Try solving on your own before revealing the answer!

Q8. (6 + 7y)(-3y + 10)

Background

Topic: Multiplying Binomials with Variables and Constants

This question tests your ability to multiply binomials with both variable and constant terms.

Key Terms and Formulas

• Distributive Property: Multiply each term in the first binomial by each term in the second binomial.

$(6 + 7y)(-3y + 10)$

Step-by-Step Guidance

1. Multiply 6 by -3y and by 10.

2. Multiply 7y by -3y and by 10.

3. Write out all four terms.

4. Combine like terms to simplify.

Try solving on your own before revealing the answer!

Q9. (x + 2)(x^2 – 3x + 4)

Background

Topic: Multiplying a Binomial by a Trinomial

This question tests your ability to multiply a binomial by a trinomial using the distributive property.

Key Terms and Formulas

• Distributive Property: Each term in the binomial is multiplied by each term in the trinomial.

$(x + 2)(x^2 - 3x + 4)$

Step-by-Step Guidance

1. Multiply x by each term in the trinomial.

2. Multiply 2 by each term in the trinomial.

3. Write out all six terms.

4. Combine like terms to simplify.

Try solving on your own before revealing the answer!

Q10. Use a graphic organizer to find the product of (-6h + 5h^2 + 4)(-3h + 2).

Background

Topic: Multiplying Polynomials (Binomial by Trinomial)

This question tests your ability to multiply a trinomial by a binomial using a graphic organizer or area model.

Key Terms and Formulas

• Distributive Property: Each term in the first polynomial is multiplied by each term in the second polynomial.

$(-6h + 5h^2 + 4)(-3h + 2)$

Step-by-Step Guidance

1. Multiply each term in the first polynomial by -3h.

2. Multiply each term in the first polynomial by 2.

3. Write out all six terms.

4. Combine like terms to simplify.

Try solving on your own before revealing the answer!

Q11. Carla is painting rectangular skyscraper backdrops. Find the area of the larger skyscraper, the smaller skyscraper, and the total area Carla will paint.

Background

Topic: Area of Rectangles (Polynomial Multiplication)

This question tests your ability to find the area of rectangles with polynomial dimensions and sum the areas.

Key Terms and Formulas

• Area of Rectangle: Multiply length by width.

• Polynomial Multiplication: Use distributive property.

For example, if dimensions are (2x + 5) and (x – 7):

$(2x + 5)(x - 7)$

Step-by-Step Guidance

1. Multiply each term in the first polynomial by each term in the second polynomial for both rectangles.

2. Write out all terms for each area.

3. Combine like terms for each area.

4. Add the two areas to find the total area.

Try solving on your own before revealing the answer!

Final Answer Examples

For (x + 4)(x + 2): $x^2 + 6x + 8$

For (x - 5)(x + 3): $x^2 - 2x - 15$

For (2x + 7)(3x - 1): $6x^2 + 19x - 7$

For (3r^2 - 2)(r - 9): $3r^3 - 27r^2 - 2r + 18$

For (-x + 11)(-5x + 8): $5x^2 - 8x - 55x + 88$

For (6 + 7y)(-3y + 10): $-18y + 60 - 21y^2 + 70y$

For (x + 2)(x^2 - 3x + 4): $x^3 - 3x^2 + 4x + 2x^2 - 6x + 8$

For (-6h + 5h^2 + 4)(-3h + 2): $18h^2 - 12h - 15h^3 + 10h^2 - 12h + 8$

For the skyscraper areas, expand and add the polynomials as shown in the steps above.

Each answer is found by multiplying each term and combining like terms.