Problem 55 - Algebra A Combined Function

Question

Simplify each polynomial by combining any like terms. See Examples 13 and 14. $$ 5 x^{2} y+6 x y^{2}-5 y x^{2}+4-9 y^{2} x $$

Step-by-Step Solution

Answer

The simplified polynomial is \(-3xy^2 + 4\).

Step 1: Identify Like Terms

In the polynomial \(5x^2y + 6xy^2 - 5yx^2 + 4 - 9y^2x\), identify terms that are like terms. Like terms have the same variables raised to the same power. So, for example, \(5x^2y\) and \(-5yx^2\) are like terms because they both contain \(x^2y\).

Step 2: Group Like Terms Together

Group the like terms together. For the given polynomial: \[(5x^2y - 5yx^2) + (6xy^2 - 9y^2x) + 4\] Notice that terms like \(-5yx^2\) can be rewritten as \(-5x^2y\) since multiplication is commutative.

Step 3: Simplify Combined Like Terms

Simplify each group of like terms. For the group \((5x^2y - 5x^2y)\), when simplified, it becomes \(0\) since \(5 - 5 = 0\). For the group \((6xy^2 - 9y^2x)\), simplify it to \(-3xy^2\), because \(6 - 9 = -3\).

Step 4: Write the Simplified Polynomial

Combine all simplified terms and write the new simplified polynomial. The simplified version is \(-3xy^2 + 4\), since \(5x^2y - 5x^2y = 0\) and does not need to be written.

Key Concepts

Like Terms

In algebra, a key concept to understand is **like terms**. Like terms are terms that have the exact same variable components, each raised to the same power. This means that the coefficient, or the numerical factor in front of the terms, can be different, but the variable parts must match exactly.
For example:

• In the expression \(5x^2y - 5yx^2 + 6xy^2\), the terms \(5x^2y\) and \(-5yx^2\) are considered like terms. Even though the order of the variables is different, they correspond to the same variable part \(x^2y\) because multiplication is commutative.
• On the other hand, \(6xy^2\) and \(-9y^2x\) might at first seem different. However, due to the commutative property, \(-9y^2x\) can be seen as \(-9xy^2\), making them also like terms.

Identifying and then combining like terms allows us to simplify polynomials efficiently by adding or subtracting the coefficients while maintaining the variable parts unchanged.

Polynomial Terms

When learning about polynomials, it's important to recognize individual **polynomial terms**. Each polynomial is made up of terms, which can consist of variables, coefficients, or constants. A polynomial is essentially a sum of such terms.
Here's how polynomial terms work:

• A term like \(5x^2y\) is a product of a coefficient (\