Question

Use the distributive property and mental math to simplify the expression. $$ 3 x^{2}+4 x+8-7 x^{2} $$

Short answer

The simplified expression is \(-4x^{2} + 4x + 8\).

Step-by-step solution

Apply the Distributive Property

The distributive property involves adding or subtracting terms that have the same variables with the same exponent. Therefore, by using distribution, the expression becomes: \(3 x^{2} - 7x^{2} + 4x + 8\).

Combine Similar Terms

Combining the similar terms results in: \(-4x^{2} + 4x + 8\)

Key concepts

Distributive Property

The distributive property is a cornerstone in simplifying algebraic expressions, serving as a bridge to unlock a variety of problems. In its essence, the distributive property allows you to multiply a single term across the terms within a parenthesis. For instance, in the expression \[ a(b+c) \], the property tells us to distribute the multiplication of \( a \) to both \( b \) and \( c \), resulting in \( ab + ac \).

Applying this property helps in unraveling expressions and setting the stage for further simplification. The exercise presented illustrates a slightly different use of the distributive property, where we are effectively distributing the subtraction across two like terms with the variable \( x^{2} \). In other words, \( 3x^2 - 7x^2 \) is an internal distribution of \( (-1) \) across the \( x^2 \) terms, which simplifies to \( -4x^2 \). This property is particularly powerful because it streamlines expressions, making them more manageable for subsequent operations.

Combining Like Terms

Moving forward with the expression simplification, we encounter the need to 'combine like terms.' Like terms are terms that have exactly the same variables raised to the same power, even if the coefficients (numbers in front) are different. In the given expression \( -4x^2 + 4x + 8 \), there are no like terms other than the \( x^2 \) terms which have been already combined using the distributive property.

We can only combine terms that are similar; for example, \( x \) terms with other \( x \) terms, or constant numbers with other constants. It's like organizing a closet—shirts with shirts, pants with pants. By keeping like terms together, we not only tidy up the expression but also prepare it for potential further operations such as factorization or solving equations.

Mental Math

The technique of 'mental math' plays a supporting role in simplifying algebraic expressions. It involves performing calculations in your head, without the aid of a calculator or paper. When applying mental math to algebra, it's about quickly identifying patterns, like terms, and simple arithmetic operations that can combine or reduce parts of the expression.

In our example \( -4x^2 + 4x + 8\), there's an opportunity to use mental math to add the numerical coefficient of \( x^2 \) terms right away, and then swiftly move to the constants. Here, there is no mental math needed for simplifying constants since there is only one constant term, \( 8 \). Developing strong mental math skills can exponentially speed up the problem-solving process, enabling students to tackle more complex challenges with greater confidence and efficiency.