Question

kh math 2a 25/26
multiplying polynomials and simplifying expre
multiplying multivariable polynomials
what is the product of $2x + y$ and $5x - y + 3$?
$\square x^2+ \square xy+ \square x+ \square y^2+ \square y$

Explanation:

Step1: Apply distributive property

Multiply \(2x + y\) with each term in \(5x - y + 3\):
\((2x + y)(5x - y + 3)=2x(5x - y + 3)+y(5x - y + 3)\)

Step2: Distribute further

For \(2x(5x - y + 3)\):
\(2x\times5x = 10x^{2}\), \(2x\times(-y)=-2xy\), \(2x\times3 = 6x\)
For \(y(5x - y + 3)\):
\(y\times5x = 5xy\), \(y\times(-y)=-y^{2}\), \(y\times3 = 3y\)

Step3: Combine like terms

• \(x^{2}\) term: \(10x^{2}\) (coefficient \(10\))
• \(xy\) terms: \(-2xy + 5xy=3xy\) (coefficient \(3\))
• \(x\) term: \(6x\) (coefficient \(6\))
• \(y^{2}\) term: \(-y^{2}\) (coefficient \(-1\))
• \(y\) term: \(3y\) (coefficient \(3\))

Answer:

The coefficients are \(10\) (for \(x^{2}\)), \(3\) (for \(xy\)), \(6\) (for \(x\)), \(-1\) (for \(y^{2}\)), and \(3\) (for \(y\)). So the filled expression is \(10x^{2}+3xy + 6x-1y^{2}+3y\) (or \(10x^{2}+3xy + 6x - y^{2}+3y\)).