Question

Multiplicar los polinomios.

A. (−3x – 4) (4x + 5)
B. (3y + 7) (y − 4)
C. (3x − 1) (2x + 3)
D. (5y + 2) (2y − 1)
E. (x + 2) (x − 3)
F. (−3x – 4) (−3x + 4)
G. (y + 4) (y^2 – 4y + 16)

Answer 1

A. When multiplying ((-3x - 4)(4x + 5)), we use the distributive property, resulting in
`\(-12x^2 - 7x - 20\)`.

B. In the multiplication of ((3y + 7)(y - 4)), applying the distributive property yields
`\(3y^2 - 5y - 28\)`.

C. For ((3x - 1)(2x + 3)), the distributive property gives
`\(6x^2 + 7x - 3\)`.

D. Multiplying ((5y + 2)(2y - 1)) using the distributive property results in
`\(10y^2 + 8y - 2\)`.

E. In the case of ((x + 2)(x - 3)), applying the distributive property yields
`\(x^2 - x - 6\)`.

F. When multiplying ((-3x - 4)(-3x + 4)), the distributive property gives
`\(9x^2 + 16\)`.

G. Finally, for
`\((y + 4)(y^2 - 4y + 16)\)`, using the distributive property results in
`\(y^3 + 12y^2 + 48y + 64\)`.

Explanation:

Polynomial multiplication is a process where each term in one polynomial is systematically multiplied by every term in another, and the results are then combined. This is accomplished using the distributive property, which dictates that each term in the first polynomial must be multiplied by each term in the second polynomial. The obtained products are then summed up, and like terms are combined.

Taking the example of ((3x - 1)(2x + 3)), this process results in the final answer of
`\(6x^2 + 7x - 3\)`. Here, the term (3x) from the first polynomial is multiplied by both (2x) and (3) from the second polynomial, and the term (-1) is similarly distributed. The products are then combined to form the final polynomial.

This method is repeated for each part of the original question, yielding expressions such as
`\(-12x^2 - 7x - 20\)` and
`\(3y^2 - 5y - 28\)`. Polynomial multiplication is a fundamental concept in algebraic manipulation, and mastering this process is crucial for solving various mathematical problems involving polynomials.