91Ó°ÊÓ

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by each term within a set of parentheses. It ensures that multiplication is equally distributed to each component of the expression. This property is expressed mathematically as:
\[a(b + c) = ab + ac\]When applying this property, it's important to remember:

• You multiply the term outside the parentheses by each of the terms inside the parentheses.
• Each resulting product is an individual term of the expression.
• The operation maintains the equality of expressions.

For example, in the expression \[-b^{2}(6b^{3} - 16b + 11)\],each term inside the parentheses is multiplied by \(-b^{2}\). By doing so, you ensure that every term in the parentheses is accounted for, allowing for the accurate expansion of the expression.

Polynomial multiplication involves multiplying variables raised to powers, following algebraic rules. When multiplying polynomials, look at each term carefully from each expression. Here's the basic approach:

• Multiply the coefficients (numerical values) together.
• Add the exponents of like bases.
• Combine the results to form a new polynomial.

In the given problem, once the distributive property has been applied, each individual multiplication is completed as follows:
\[-b^{2} imes 6b^{3} = -6b^{5}\],which arises by multiplying the coefficients (-1 and 6) and adding the exponents of b (2 and 3).Also,
\[-b^{2} imes -16b = 16b^{3}\],where coefficients multiply to give 16 and the exponents add (2 + 1),and
\[-b^{2} \times 11 = -11b^{2}\], keeping the base b's exponent the same. As these calculations show, attention to detail in each step is essential to achieve the correct result.

Simplifying expressions is the final step in most algebra problems. After applying the distributive property and performing the necessary multiplications, it's crucial to simplify the expression to its simplest form. The key points are:

• Combine like terms, which are terms with the same variable raised to the same power.
• Write the terms in descending order of the powers.
• Ensure there are no further possible simplifications.

The expression\[-6b^{5} + 16b^{3} - 11b^{2}\]is already simplified. It contains no like terms to combine and is arranged neatly by the exponents in descending order. In simple terms, the goal of simplifying is to make an expression as compact and straightforward as possible without altering its meaning.