91Ó°ÊÓ

The distributive property is a fundamental concept in algebra. It allows us to multiply a single term by each term within a polynomial. In essence, we 'distribute' the multiplication across each term in the polynomial this way:
For example, if you have the polynomial expressions \(a \times (b + c)\), you distribute as follows:

• \(a \times b\)
• \(a \times c\)

Combining them, we get \(a \times (b + c) = ab + ac\).

In our exercise, we applied the distributive property twice:

• First, \(4z\) was multiplied by each term in \(2z^3 - 5z^2 + 8z - 1\).
• Next, we multiplied \(3\) by each term in \(2z^3 - 5z^2 + 8z - 1\).
  

This resulted in two separate sets of terms which we then combined in the next step.

Combining like terms involves grouping terms that have the same variable raised to the same power. This simplification is a crucial step in polynomial operations.

In our exercise, after distributing, we obtained:
\(8z^4 - 20z^3 + 32z^2 - 4z\)
and
\(6z^3 - 15z^2 + 24z - 3\).

• We then added these groups of terms together: \[(8z^4 - 20z^3 + 32z^2 - 4z) + (6z^3 - 15z^2 + 24z - 3)\]
• To combine like terms, we added the coefficients of terms with the same powers of \(z\):
  

• \(8z^4\)
• \(-20z^3 + 6z^3 = -14z^3\)
• \(32z^2 - 15z^2 = 17z^2\)
• \(-4z + 24z = 20z\)
• \(-3\) remains the same.
  

Hence, we simplified the expression to: \(8z^4 - 14z^3 + 17z^2 + 20z - 3\).Combining like terms is key because it makes the equation simpler and easier to understand.

Algebraic expressions are combinations of variables, numbers, and operators (such as addition and multiplication). These expressions often contain terms where a term is a product of a number (coefficient) and a variable raised to an exponent.
For example, in the expression \(2z^3 - 5z^2 + 8z - 1\):

• \(2z^3\) is a term where \(2\) is the coefficient and \(z^3\) is the variable part.
• \(-5z^2\), \(8z\), and \(-1\) are other terms.
  

When multiplying polynomials, each term in one polynomial must be multiplied by each term in the other polynomial, using the distributive property.

Combining all the generated terms (and ensuring to combine like terms correctly) results in a new polynomial, which is an algebraic expression representing the product of the original polynomials. Mastery of these concepts allows students to solve more complex algebraic problems with confidence.