Problem 105 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ (x-7)\left(x^{2}+x-3\right) $$

Step-by-Step Solution

Answer

Question: Multiply the expressions (x - 7) and (x^2 + x - 3), and simplify the result. Answer: The product of the given expressions, when simplified, is x^3 - 6x^2 - 10x + 21.

Step 1: Distribute the first term

Multiply the first term, x, of the first expression with each term in the second expression: $$ x(x^2) + x(x) + x(-3) $$ This simplifies to: $$ x^3 + x^2 - 3x $$

Step 2: Distribute the second term

Multiply the second term, -7, of the first expression with each term in the second expression: $$ -7(x^2) + -7(x) + -7(-3) $$ This simplifies to: $$ -7x^2 - 7x + 21 $$

Step 3: Combine the two expressions

Now, combine the two expressions obtained in Step 1 and Step 2: $$ (x^3 + x^2 - 3x) + (-7x^2 - 7x + 21) $$

Step 4: Combine like terms

Finally, combine any like terms in the expression: $$ x^3 + x^2 - 7x^2 -3x - 7x + 21 $$ This simplifies to: $$ x^3 - 6x^2 - 10x + 21 $$ The final answer is: $$ x^3 - 6x^2 - 10x + 21 $$

Key Concepts

Combining Like Terms

When you come across a problem that involves multiplying polynomials as in the above exercise, it's essential to know how to combine like terms correctly. Like terms are terms that have the same variables raised to the same power. For instance, in the expression \(x^2 - 7x^2\), both terms have the same variable, \(x\), raised to the same power of 2. To combine them, you just add or subtract the coefficients (the numbers in front of the variables), depending on their signs.

To effectively combine like terms:

• First, identify the like terms.
• Add or subtract the coefficients while keeping the variable and its exponent unchanged.
• Repeat this process for all like terms in the expression.

By the end, you should have a simplified expression with no repeated variable terms. In the given exercise, the terms \(x^2\) and \( - 7x^2\) were combined to result in \( - 6x^2\), and \( - 3x\) and \( - 7x\) were combined to \( - 10x\).

Distributive Property

The distributive property is a cornerstone of algebra and is used to multiply a single term by each term of a polynomial. This property is crucial when dealing with the multiplication of polynomials.

The distributive property states that for any three addends, say \(a\), \(b\), and \(c\), the equality \(a(b + c) = ab + ac\) holds true. This means that to multiply \(a\) by the sum of \(b\) and \(c\), one can multiply \(a\) by \(b\) and \(a\) by \(c\), and then sum the results. Applying this to the given problem, we multiplied \(x\) by each term in the trinomial \(x^2 + x - 3\) first and then \( -7\) by each term.

Visualizing the Distributive Property

Imagine each term in one polynomial as a stretching hand to distribute itself to each term in the other polynomial. This visualization helps ensure that no terms are forgotten during the multiplication steps.

Algebraic Expression Simplification

The final stage of solving polynomial multiplication problems, such as the one presented, is to simplify the algebraic expression. Algebraic expression simplification aims to make expressions easier to understand and work with by reducing them to their simplest form.

The process involves several steps, beginning with the distribution of terms, as mentioned previously, and followed by combining like terms to condense the expression. Once the expression doesn't have any more like terms to combine, you should rearrange the remaining terms in descending order of their powers. This allows for a standardized form and often makes it easier to compare with other polynomials or to carry out further mathematical operations.

In the provided exercise, the final simplified form is \(x^3 - 6x^2 - 10x + 21\). It is written in descending order, and no terms can be further combined. The simplification makes the polynomial much more manageable for further analysis or graphing.