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The distributive property is an essential building block in algebra. It allows you to simplify expressions by distributing one term across other terms inside parentheses. When you see an expression like \(a(b + c)\), the distributive property tells us to multiply \(a\) by both \(b\) and \(c\), resulting in \(ab + ac\).
In our exercise, we use the distributive property twice:
First, we distribute the term \(p\) from \( (p - 4)\) across all terms in \(p^2 - 6p + 9\). This gives us \(p \cdot p^2 = p^3\), \(p \cdot (-6p) = -6p^2\), and \(p \cdot 9 = 9p\).
Next, we distribute the term \(-4\) from \( (p - 4)\). We get \(-4 \cdot p^2 = -4p^2\), \(-4 \cdot (-6p) = 24p\), and \(-4 \cdot 9 = -36\).
The distributive property helps us break down complex expressions into simpler parts, making it easier to perform further operations.

Multiplying binomials with polynomials might seem tricky at first, but it's just an extension of the distributive property. A binomial is an algebraic expression with two terms, like \(p - 4\). A polynomial has more than two terms, like \(p^2 - 6p + 9\). When multiplying them, we need to ensure each term in the binomial gets multiplied by every term in the polynomial.
In our example \((p - 4)(p^2 - 6p + 9)\), we first focus on the \(p\) in \(p - 4\):

• \(p \cdot p^2 = p^3\)
• \(p \cdot (-6p) = -6p^2\)
• \(p \cdot 9 = 9p\)

This distributes \p\ across all terms in the polynomial.
Now, we handle the second term \(-4\):

• \(-4 \cdot p^2 = -4p^2\)
• \(-4 \cdot (-6p) = 24p\)
• \(-4 \cdot 9 = -36\)

After performing these multiplications, we move on to combine the results.

Once you've distributed all terms, the next step is to simplify the expression by combining like terms. Like terms are terms that have the same variable raised to the same power. For instance, \(9p\) and \(24p\) are like terms because both contain the variable \(p\) to the power of 1.
For our exercise, after distribution, we have: \(p^3\), \(-6p^2\), \(-4p^2\), \(9p\), \(24p\), and \(-36\).
We combine these like terms as follows:

• Combine \(-6p^2\) and \(-4p^2\): \(-6p^2 - 4p^2 = -10p^2\)
• Combine \(9p\) and \(24p\): \(9p + 24p = 33p\)

The final, simplified expression is then: \(p^3 - 10p^2 + 33p - 36\).
Combining like terms helps us tidy up the expression and makes it easier to understand and use in further calculations.