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The distributive property is a fundamental concept in algebra that helps us simplify expressions involving multiplication over addition or subtraction. It allows us to distribute a scalar across the terms inside a bracket, which often makes the expression easier to manipulate. For instance, if you have an expression like \( a(b + c) \), it can be expanded to \( ab + ac \).
In our exercise, we apply the distributive property by multiplying 1.7 with the terms inside the second bracket. This calculation involves performing the operation \( 1.7 \times (6x^3 - 4x^2) \).
This step is crucial because it helps us break down more complex expressions into simpler components, ready for further simplification.

Once the distributive property has been applied, the next essential step is combining like terms. Like terms are terms in an expression that have the same variables raised to the same power. For example, \( 3x^2 \) and \( 5x^2 \) are like terms because they both contain \( x^2 \).
This step involves adding or subtracting the coefficients of like terms to consolidate them into one term.

• In our expression \( 6x^3 - 10.2x^3 - 16x^2 + 6.8x^2 \), you identify the like terms \( 6x^3 \) and \( -10.2x^3 \), as well as \( -16x^2 \) and \( 6.8x^2 \).
• Add or subtract their coefficients: \( 6 - 10.2 = -4.2 \) gives you \( -4.2x^3 \) and \( -16 + 6.8 = -9.2 \) gives you \( -9.2x^2 \).

Combining like terms simplifies the expression further and brings you closer to the final simplified form.

Polynomial simplification involves using various algebraic techniques to express polynomials in a simpler form. In this exercise, we've already seen how applying the distributive property and combining like terms simplifies a polynomial.
Let’s consider our expression: \( -4.2x^3 - 9.2x^2 \). After using the distributive property and combining like terms, our polynomial is now in its simplest form.
When simplifying polynomials, always ensure to:

• Identify and group like terms correctly.
• Perform accurate arithmetic operations to combine these terms.
• Repeatedly use the distributive property when necessary, especially for expressions with more than one bracket or factor.

These methods help to reorganize and simplify components without altering the value of the original expression. Applying these techniques enhances your algebraic manipulation skills, streamlining complex expressions into manageable forms.