91Ó°ÊÓ

The distributive property is a fundamental concept in algebra. It allows you to simplify expressions by distributing a number or variable across terms inside parentheses. For example, in the expression \( a(b + c) \), you apply the distributive property by multiplying \( a \) with both \( b \) and \( c \), resulting in \( ab + ac \). In the exercise, we apply the distributive property to the term \( 3(t - 2t^3) \).

Distributing the 3, we get:
\( 3 \times t - 3 \times 2t^3 = 3t - 6t^3 \).
This step helps to break down the initial expression into simpler parts that are easier to manage and combine later.

Always pay attention to the sign in front of the term you are distributing. It can change the resulting expression significantly.

Combining like terms is a crucial skill in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. For example, \( 5t^3 \) and \( -6t^3 \) are like terms because they both contain \( t^3 \).

In the exercise, once we distribute the 3, we are left with:

• \( 5t^3 \)
• \( t \)
• \( 3t \)
• \( -6t^3 \)

Combine the like terms: The terms \( 5t^3 \) and \( -6t^3 \) are combined as follows:

• \( 5t^3 - 6t^3 = -t^3 \)

Similarly, the terms \( t \) and \( 3t \) are combined:

• \( t + 3t = 4t \)

Putting them together, we get the simplified expression:
\( -t^3 + 4t \).

Remember that combining like terms involves adding or subtracting the coefficients while keeping the variable part unchanged.

Polynomials are algebraic expressions that consist of variables and coefficients. They are made up of terms that can be added, subtracted, multiplied, or divided (except by the variable). Each term in a polynomial contains a variable raised to a non-negative integer exponent.

In the given exercise, the expression \( 5t^3 + t + 3(t - 2t^3) \) is a polynomial.
The terms in this polynomial are:

• \( 5t^3 \)
• \( t \)
• \( 3(t - 2t^3) \)

After applying the distributive property and combining like terms, the simplified result is also a polynomial: \( -t^3 + 4t \).

Polynomials can have different numbers of terms, and they are classified based on their degree, which is the highest power of the variable in the polynomial. In this case, the degree is 3, as the highest power of \( t \) is \( t^3 \).

Understanding polynomials is key to mastering many algebraic operations.