Question

Challenge Problems. $$3 a^{2} x+5 a^{3} x^{3}-2 a x^{2} b y\left(-a^{2} x^{2}\right)$$

Short answer

The simplified expression is \(3a^2x + 5a^3x^3 + 2a^3bx^4y\).

Step-by-step solution

Identify the terms

The expression provided is a polynomial with four terms: \(3a^2x\), \(5a^3x^3\), \(-2ax^2by\), and \((-a^2x^2)\). The last term is in parentheses and multiplies the third term.

Distribute the negative sign

Distribute the negative sign through the parentheses. To do this, we need to multiply \(-2ax^2by\) by \(-a^2x^2\). The negative sign will make the multiplication positive, and we need to multiply the coefficients and add the exponents of same base.

Perform the multiplication

Multiply the coefficients \(-2\) and \(-1\) to get a positive \(2\). Multiply the variables by adding the exponents: \(x^2\) and \(x^2\) become \(x^{2+2} = x^4\), \(a\) and \(a^2\) become \(a^{1+2} = a^3\). The term \(b\) and \(y\) remains unchanged because they have no like terms to combine with in this step.

Simplify the expression

Combine the multiplication result with the remaining terms: \(3a^2x + 5a^3x^3 + 2a^3bx^4y\). This is the simplified form of the given polynomial, as there are no like terms to combine.

Key concepts

Distributive Property

The distributive property is a pillar of algebra that allows us to simplify complex expressions. In essence, it states that multiplication is distributive over addition and subtraction. This means that a single term can be distributed, or multiplied, across the terms inside a parenthesis.

For example, if we have an expression like \( b(a + c) \), the distributive property tells us to multiply \( b \) by both \( a \) and \( c \) individually. So, it results in \( ba + bc \). Similarly, if we encounter negative signs, such as in the expression \( -x(y - z) \), we distribute \( -x \) to obtain \( -xy + xz \). This action often simplifies expressions significantly, preparing them for further simplification through other means, like combining like terms.

Combining Like Terms

When simplifying polynomials, it's essential to combine like terms to reduce the expression to its simplest form. Like terms are terms that have the exact same variables raised to the same power. Coefficients may vary, but as long as the variable part is identical, terms can be combined.

Let's look at an example with the expression \( 5x^2 + 3x - 2x^2 + x \). Here, \( 5x^2 \) and \( -2x^2 \) are like terms because both contain the variable \( x \) raised to the second power. We can combine them by adding their coefficients to get \( 3x^2 \). Similarly, \( 3x \) and \( x \) can be combined as \( 4x \) since they both contain the variable \( x \) to the first power. Combining like terms makes understanding and solving polynomial equations much easier.

Exponent Rules

Exponent rules, or laws of exponents, are guidelines for handling powers of numbers or variables. These rules simplify the process of working with exponents in algebraic expressions.

The most relevant exponent rules for polynomial simplification are:

• Product of Powers: \( a^m \cdot a^n = a^{m+n} \), which tells us to add exponents when multiplying like bases.
• Power of a Power: \( (a^m)^n = a^{m \cdot n} \), which says to multiply exponents when raising a power to a power.
• Power of a Product: \( (ab)^n = a^n \cdot b^n \), implying that when a product is raised to an exponent, we distribute the exponent to both bases.

For instance, in the expression \( x^2 \cdot x^3 \), by applying the product of powers rule, we combine the exponents to get \( x^{2+3} = x^5 \). Understanding these rules is key to correctly consolidating terms in polynomial equations.