Problem 96 - Elementary Algebra

Question

Problem: Simplify: \((2 p-9)\left(3 p^{2}+6 p-5\right)\) Incorrect Answer: $$ \begin{aligned} &(2 p-9)\left(3 p^{2}+6 p-5\right) \\ &=2 p\left(3 p^{2}\right)+2 p(6 p)+2 p(-5)-9\left(3 p^{2}\right)-9(6 p)-9(-5) \\\ &=6 p^{3}+12 p^{2}+10 p-27 p^{2}-54 p+45 \\ &=6 p^{3}-15 p^{2}-44 p+45 \end{aligned} $$

Step-by-Step Solution

Answer

The simplified expression is \(6p^{3} - 15p^{2} - 64p + 45\).

Step 1: Apply Distribution

Start by using the distributive property to expand the expression \( (2p-9)(3p^{2}+6p-5) \). Multiply each term in \(2p-9\) by each term in \(3p^{2}+6p-5\).

Step 2: Multiply 2p by Each Term

Multiply \(2p\) by each term in the polynomial \(3p^{2} + 6p - 5\): \(2p \times 3p^{2} = 6p^{3} \) \(2p \times 6p = 12p^{2} \) \(2p \times (-5) = -10p \)

Step 3: Multiply -9 by Each Term

Next, multiply \(-9\) by each term in the polynomial \(3p^{2} + 6p - 5\): \(-9 \times 3p^{2} = -27p^{2} \)\(-9 \times 6p = -54p \)\(-9 \times (-5) = 45 \)

Step 4: Combine Like Terms

Now, combine all the terms from the previous steps: \(6p^{3} + 12p^{2} - 10p - 27p^{2} - 54p + 45\).Combine the like terms: \(6p^{3} + (12p^{2} - 27p^{2}) + (-10p - 54p) + 45\). Simplify: \(6p^{3} - 15p^{2} - 64p + 45\)

Step 5: Final Answer

The simplified expression is: \(6p^{3} - 15p^{2} - 64p + 45\)

Key Concepts

distributive property

The Distributive Property is a key concept in algebra that helps us simplify expressions. It states that multiplying a sum (or difference) by a number is the same as multiplying each term inside the sum (or difference) by that number and then adding (or subtracting) the results.
For example, consider the expression \[ a(b + c) \]. Using the distributive property, we get \[ ab + ac \].
In our exercise, we apply this property to \((2p-9)(3p^2 + 6p - 5)\). We must multiply each term in \[2p-9\] by each term in \[3p^2 + 6p - 5\]. This means:

• Multiply \[2p\] by \[3p^2\], \[6p\], and \[-5\]
• Multiply \[-9\] by \[3p^2\], \[6p\], and \[-5\]

This step ensures that each term of the polynomial is expanded correctly.

combine like terms

Combining like terms is an essential part of simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. For example, \[6p^2\] and \[-27p^2\] are like terms because both terms contain \[p^2\].
After applying the distributive property in our exercise, we obtain:
\[6p^3 + 12p^2 -10p -27p^2 -54p + 45\]
The next step involves merging these like terms. We identify terms with the same power and add or subtract their coefficients:

• Combine \[12p^2\] and \[-27p^2\] as follows: \[12p^2 - 27p^2 = -15p^2\]
• Simplify \[-10p\] and \[-54p\]: \[-10p - 54p = -64p\]

Now, we have a simplified expression with combined like terms: \[6p^3 - 15p^2 - 64p + 45\].

simplify expressions

Simplifying expressions makes them easier to understand and work with. It involves applying various algebraic rules and operations systematically.
Here's how we simplify our given exercise:

• Apply the distributive property to expand \((2p-9)(3p^2 + 6p - 5)\).
• Multiply each term in the polynomial by each term in \[2p - 9\], using step-by-step multiplication.
• Combine like terms by identifying and merging terms with the same variables and powers.

Now, reduce to its simplified form:
Starting with:
\[6p^3 + 12p^2 - 10p - 27p^2 - 54p + 45\]
Combining like terms:
\[6p^3 - 15p^2 - 64p + 45\]
This process gives us a cleaner version of the expression, \[6p^3 - 15p^2 - 64p + 45\], which is easier to work with for further calculations or understanding.