Question

distributive property - multiplying
example: $3x^2(2x^4 - 5)$
multiplying monomials by polynomials
$2x(6x^4 + x^3)$

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

you try:

1. $-3x^4(8x^4 - 2x^7)$
2. $5y^2(4x^3y + x^2y - 2xy^3)$

Explanation:

Step1: Multiply the monomial by each term in the polynomial

For the problem \(-3x^{4}(8x^{4}-2x^{7})\), we use the distributive property \(a(b - c)=ab - ac\), where \(a = -3x^{4}\), \(b = 8x^{4}\), and \(c = 2x^{7}\).
First, multiply \(-3x^{4}\) by \(8x^{4}\): \(-3x^{4}\times8x^{4}=-24x^{4 + 4}=-24x^{8}\) (using the rule \(x^{m}\times x^{n}=x^{m + n}\)).
Then, multiply \(-3x^{4}\) by \(-2x^{7}\): \(-3x^{4}\times(-2x^{7}) = 6x^{4+7}=6x^{11}\) (using the same exponent rule and the rule that the product of two negative numbers is positive).

Step2: Combine the results

Putting it together, we get \(-24x^{8}+6x^{11}\) (or \(6x^{11}-24x^{8}\) by rearranging in descending order of exponents).

Answer:

\(6x^{11}-24x^{8}\) (or \(-24x^{8}+6x^{11}\))