Simplify (2xy^2)(x^2-5x+3y)

Solution:

Step 1

Utilize the distributive property to expand the expression: $2xy^2(x^2) + 2xy^2(-5x) + 2xy^2(3y)$.

Step 2

Begin simplification.

Step 2.1

Combine like terms by multiplying $x$ and $x^2$.

Step 2.1.1

Rearrange to show multiplication: $2(x^2 \cdot x)y^2 + 2xy^2(-5x) + 2xy^2(3y)$.

Step 2.1.2

Execute the multiplication of $x^2$ and $x$.

Step 2.1.2.1

Express $x$ as $x^1$: $2(x^2 \cdot x^1)y^2 + 2xy^2(-5x) + 2xy^2(3y)$.

Step 2.1.2.2

Apply the exponent rule $a^m \cdot a^n = a^{m+n}$: $2x^{2+1}y^2 + 2xy^2(-5x) + 2xy^2(3y)$.

Step 2.1.3

Add the exponents $2$ and $1$: $2x^3y^2 + 2xy^2(-5x) + 2xy^2(3y)$.

Step 2.2

Multiply $x$ by itself.

Step 2.2.1

Reposition $x$: $2x^3y^2 + 2(x \cdot x)y^2(-5) + 2xy^2(3y)$.

Step 2.2.2

Carry out the multiplication of $x$ and $x$: $2x^3y^2 + 2x^2y^2(-5) + 2xy^2(3y)$.

Step 2.3

Multiply $y^2$ by $y$.

Step 2.3.1

Reposition $y$: $2x^3y^2 + 2x^2y^2(-5) + 2x(y \cdot y^2)(3)$.

Step 2.3.2

Perform the multiplication of $y$ and $y^2$.

Step 2.3.2.1

Express $y$ as $y^1$: $2x^3y^2 + 2x^2y^2(-5) + 2x(y^1 \cdot y^2)(3)$.

Step 2.3.2.2

Apply the exponent rule $a^m \cdot a^n = a^{m+n}$: $2x^3y^2 + 2x^2y^2(-5) + 2xy^{1+2}(3)$.

Step 2.3.3

Sum the exponents $1$ and $2$: $2x^3y^2 + 2x^2y^2(-5) + 2xy^3(3)$.

Step 3

Finalize the simplification of each term.

Step 3.1

Multiply $-5$ by $2$: $2x^3y^2 - 10x^2y^2 + 2xy^3(3)$.

Step 3.2

Multiply $3$ by $2$: $2x^3y^2 - 10x^2y^2 + 6xy^3$.

Knowledge Notes:

The problem involves simplifying a polynomial expression by applying algebraic rules and properties. The key points and relevant knowledge include:

1. Distributive Property: This property allows us to multiply a single term by each term within a parenthesis. It is expressed as $a(b + c) = ab + ac$.

2. Combining Like Terms: This involves adding or subtracting terms that have the same variables raised to the same powers.

3. Exponent Rules: The power rule for exponents states that when multiplying two powers with the same base, you add the exponents: $a^m \cdot a^n = a^{m+n}$.

4. Multiplication of Variables: When variables are multiplied, their exponents are added if they have the same base.

5. Simplification: This is the process of rewriting an expression in a more concise and simplified form without changing its value.

In the provided solution, these concepts are applied step by step to simplify the given expression. Each step is clearly outlined, showing the progression from the original problem to the simplified result.