Problem 43 - College Algebra

Question

For the following exercises, multiply the polynomials. $$ \left(3 p^{2}+2 p-10\right)(p-1) $$

Step-by-Step Solution

Answer

The product is \(3p^3 - p^2 - 12p + 10\).

Step 1: Distribute Each Term of First Polynomial

We have to multiply each term of the polynomial \(3p^2 + 2p - 10\) by each term in the polynomial \(p - 1\). This means we'll do three separate multiplications:1. \(3p^2 \cdot (p - 1)\)2. \(2p \cdot (p - 1)\)3. \(-10 \cdot (p - 1)\)

Step 2: Multiply the First Term

Multiply the first term \(3p^2\) by each term of the second polynomial:- \(3p^2 \cdot p = 3p^3\)- \(3p^2 \cdot (-1) = -3p^2\)

Step 3: Multiply the Second Term

Repeat the process for the second term \(2p\):- \(2p \cdot p = 2p^2\)- \(2p \cdot (-1) = -2p\)

Step 4: Multiply the Third Term

Finally, multiply the third term \(-10\):- \(-10 \cdot p = -10p\)- \(-10 \cdot (-1) = 10\)

Step 5: Combine All Results

Now, combine all the products obtained:\[3p^3 - 3p^2 + 2p^2 - 2p - 10p + 10\]

Step 6: Simplify the Expression

Combine like terms in the polynomial:- Combine \(-3p^2 + 2p^2\) to get \(-p^2\).- Combine \(-2p - 10p\) to get \(-12p\).Thus, the final expression simplifies to:\[3p^3 - p^2 - 12p + 10\]

Key Concepts

Distributive Property

When you hear the term 'Distributive Property,' think of it as a way to share each term inside one math expression with every term in another. It's like passing out candies, making sure that each person gets some. In mathematics, this property helps us multiply polynomials easily. For instance, in our exercise, we use the distributive property to take each part of \[ 3p^2 + 2p - 10 \] and multiply it by every term of \( p - 1 \). This systematic approach ensures no term is left out of the multiplication process. It breaks down a complicated problem into manageable pieces. Remember:

• Every term gets its turn to multiply with all terms in the other polynomial.
• Keep track of positive and negative signs; they're just as important as the numbers.

Combining Like Terms

After distributing all the terms, you'll notice that some terms look similar. These are called 'like terms.' Combining like terms is like organizing your closet. You group all your t-shirts together and all your socks together. In polynomial terms, it means grouping similar powers of a variable together. For example:

• Combine \(-3p^2\) and \(2p^2\)
• Combine \(-2p\) and \(-10p\)

Doing this makes our expression simpler and easier to read. This step is crucial because it brings the polynomial back to a clearer form, allowing us to see the solution as\[ 3p^3 - p^2 - 12p + 10 \] in the end.

Algebraic Expressions

Algebraic expressions are the backbone of algebra. Imagine them as sentences in a book, where numbers and variables come together to form meaningful equations. In our exercise, expressions like \( 3p^2 + 2p - 10 \) are sets of these mystic sentences. An algebraic expression can contain:

• Numbers: Which tell us how many or how much.
• Variables: Denoted usually by letters, they represent unknown quantities.
• Operations: Such as addition or subtraction, dictate how the numbers and variables interact.

To simplify a moment in this vast language of expressions, understanding how these parts fit is essential. This skill is key to decoding complex polynomial problems by breaking them down into smaller, understandable expressions.

Simplifying Polynomials

Simplifying a polynomial is like tidying up a messy desk. The goal is to clean it up so everything is easy to find. Once we've finished distributing and combining like terms, simplifying is the final touch to make our polynomial look neat. This involves:

• Eliminating unnecessary terms.
• Combining like terms to reduce clutter.

The final expression \( 3p^3 - p^2 - 12p + 10 \) is a streamlined version of our original multiplication. Simplification ensures the expression is in its simplest form, making further calculations or understanding of the polynomial much clearer. This process not only makes solving math problems easier, it also helps solidify the foundational skills you need for tackling more complicated algebraic operations.