Variables and Algebraic Expressions: Number of terms

To solve this problem, we'll follow these steps:

• Step 1: Identify like terms.
• Step 2: Simplify the expression by combining like terms.

Let's simplify the given expression:
The expression is given as $x \times y - 3x \times x + 2y \times y - 6x^2$.

First, simplify each term:
- $x \times y$ remains $xy$.
- $3x \times x$ simplifies to $3x^2$.
- So, the expression becomes $xy - 3x^2 + 2y^2 - 6x^2$.

Next, combine like terms:
- Combine the $x^2$ terms: $-3x^2 - 6x^2 = -9x^2$.
- The $xy$ term and $2y^2$ term are unique and remain as they are.

This results in the simplified expression: $xy - 9x^2 + 2y^2$.

Upon comparing this simplification with the answer choices, the correct option is:

$xy - 9x^2 + 2y^2$

Therefore, the solution to the problem is $xy - 9x^2 + 2y^2$.

To solve this problem, we need to simplify the algebraic expression by performing the multiplications and then combining like terms.

Given expression:
$6x \times 3y + 4x \times x - 5y \times 2y + 2x(-9y)$

Step 1: Simplify each term by performing the multiplications

• First term: $6x \times 3y = (6 \times 3)(x \times y) = 18xy$
• Second term: $4x \times x = 4x^2$
• Third term: $5y \times 2y = (5 \times 2)(y \times y) = 10y^2$
• Fourth term: $2x(-9y) = (2 \times -9)(x \times y) = -18xy$

Step 2: Rewrite the expression with simplified terms
$18xy + 4x^2 - 10y^2 + (-18xy)$
which becomes:
$18xy + 4x^2 - 10y^2 - 18xy$

Step 3: Identify and combine like terms
Like terms are terms that have the same variable parts raised to the same powers.

• Terms with $xy$: $18xy - 18xy = 0$
• Terms with $x^2$: $4x^2$
• Terms with $y^2$: $-10y^2$

Step 4: Write the final simplified expression
After combining like terms, we get:
$4x^2 - 10y^2$

Therefore, the simplified expression is $4x^2 - 10y^2$, which corresponds to choice 4.

We'll use the law of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

We'll apply this law to the expression in the problem:

$2x^3\cdot x^2-3x\cdot x^4+6x\cdot x^2-7x^3\cdot 5=2x^{3+2}-3x^{1+4}+6x^{1+2}-35x^3$

When we apply the above law to the first three terms from the left, while remembering that any number can always be considered as that number raised to the power of 1:

$a=a^1$

And in the last term we performed the numerical multiplication,

We'll continue and simplify the expression we got in the last step:

$2x^{3+2}-3x^{1+4}+6x^{1+2}-35x^3=2x^5-3x^5+6x^3-35x^3=-x^5-29x^3$

Where in the first stage we simplified the expressions in the exponents of the terms in the expression and in the second stage we combined like terms,

Therefore the correct answer is answer A.