Question

In 9th grade terms

1. How does multiplying polynomials compare to multiplying integers?
3. Mercedes states that when multiplying 4x^3 (x^3 + 2x^2 - 3) the product is 4x^9 + 8x^6 + 8x^6 - 12x^3. What was Mercedes's error?

5. -2x^3(3x^2 - 4x + 7)
6. (2x + 6)(x-4)
8. (5y - 2)(4y^2 + 3y - 1)
9. (3x^2 + 2x - 5)( 2x - 3)

Answer 1

1. How multiplying polynomials compares to multiplying integers (9th-grade terms)
- Both use distributive ideas: when you multiply numbers you break one number into parts and multiply each part, then add the results; with polynomials you multiply each term of one polynomial by each term of the other and then add (combine) like terms.
- For integers you multiply digit values and add place-values; for polynomials you multiply coefficients and add exponents on the same variable (because x^a · x^b = x^(a+b)).
- Rules like commutative, associative, and distributive hold for both. A special case of polynomial multiplication is FOIL (First-Outer-Inner-Last) for two binomials.

3. Mercedes’s error
- She multiplied the exponents instead of adding them. For example 4x^3 · x^3 should be 4x^(3+3) = 4x^6, not 4x^9. Also 4x^3 · 2x^2 = 8x^(3+2) = 8x^5 (not 8x^6). The correct product is:
4x^3(x^3 + 2x^2 - 3) = 4x^6 + 8x^5 - 12x^3.

Problems (multiply and simplify)

5. -2x^3(3x^2 - 4x + 7)
- Multiply each term: -2x^3·3x^2 = -6x^5; -2x^3·(-4x) = +8x^4; -2x^3·7 = -14x^3.
- Answer: -6x^5 + 8x^4 - 14x^3

6. (2x + 6)(x - 4)
- FOIL: 2x·x = 2x^2; 2x·(-4) = -8x; 6·x = 6x; 6·(-4) = -24. Combine -8x+6x = -2x.
- Answer: 2x^2 - 2x - 24

8. (5y - 2)(4y^2 + 3y - 1)
- Multiply each term: 5y·4y^2 = 20y^3; 5y·3y = 15y^2; 5y·(-1) = -5y; -2·4y^2 = -8y^2; -2·3y = -6y; -2·(-1) = 2.
- Combine like terms: 20y^3 + (15y^2 - 8y^2) + (-5y - 6y) + 2 = 20y^3 + 7y^2 - 11y + 2

9. (3x^2 + 2x - 5)(2x - 3)
- Multiply: 3x^2·2x = 6x^3; 3x^2·(-3) = -9x^2; 2x·2x = 4x^2; 2x·(-3) = -6x; -5·2x = -10x; -5·(-3) = 15.
- Combine like terms: 6x^3 + (-9x^2+4x^2) + (-6x-10x) + 15 = 6x^3 - 5x^2 - 16x + 15