Problem 73 - Algebra and Trigonometry

Question

Perform the indicated operations, and simplify. \((2 x-5)\left(x^{2}-x+1\right)\)

Step-by-Step Solution

Answer

The expression simplifies to \(2x^3 - 7x^2 + 7x - 5\).

Step 1: Distribute First Term

Distribute the first term in the binomial, \(2x\), across all the terms in the trinomial \((x^2 - x + 1)\). This gives:\[2x imes x^{2} = 2x^{3}, \quad 2x imes (-x) = -2x^{2}, \quad 2x imes 1 = 2x\]So the expression becomes: \[2x^3 - 2x^2 + 2x\].

Step 2: Distribute Second Term

Now, distribute the second term in the binomial, \(-5\), across all the terms in the trinomial \((x^2 - x + 1)\). This gives:\[-5 imes x^{2} = -5x^{2}, \quad -5 imes (-x) = 5x, \quad -5 imes 1 = -5\]So these terms are: \[-5x^2 + 5x - 5\].

Step 3: Combine Like Terms

Now combine all the like terms from the two distributed polynomials:\[2x^3 - 2x^2 + 2x + (-5x^2) + 5x - 5\]Combine the terms: \[2x^3 + (-2x^2 - 5x^2) + (2x + 5x) - 5\]This simplifies to:\[2x^3 - 7x^2 + 7x - 5\].

Step 4: Final Simplification

Confirm there are no more like terms and the expression is fully simplified. The original expression has been transformed into:\[2x^3 - 7x^2 + 7x - 5\].

Key Concepts

Binomial Distribution

In polynomial mathematics, distribution is a fundamental process used when multiplying a binomial by another polynomial, such as a trinomial. Here, a binomial is an algebraic expression containing two terms connected by a plus or minus sign, for example,

• A basic binomial: \( (a + b) \)
• A trinomial: \( (x^2 - x + 1) \)

When you multiply a binomial by a trinomial, as in the exercise \((2x-5)\left(x^{2}-x+1\right)\), you distribute each term in the binomial across each term in the trinomial.
This means you:

• Multiply the first term of the binomial by each term of the trinomial.
• Repeat the same process for the second term of the binomial.

Remember, distribution helps extend expressions from smaller to larger forms, allowing simplification through combining similar or 'like' terms later.

Trinomial

A trinomial is an algebraic expression composed of three terms. For example, in the expression \(x^2 - x + 1\), each term needs to maintain its identity during the multiplication process until distribution is complete.
Here are the steps involved in managing a trinomial during multiplication:

• Identify all terms within the trinomial, such as \(x^2\), \(-x\), and \(+1\).
• Ensure each term in the trinomial is multiplied by every term in the binomial (or other polynomial forms it is multiplied with).
• Maintain signs (positive/negative) while multiplying to ensure the integrity of each term.

Trinomials are versatile in polynomial operations and often facilitate transformations through their terms, which are managed individually until combining like terms at the final steps.

Combining Like Terms

Combining like terms is a key step in simplifying algebraic expressions.
It involves consolidating terms that share the same variable and exponent. In the problem \(2x^3 - 7x^2 + 7x - 5\), the term "like terms" refers to those terms containing the same power of \(x\).
Here's how you combine like terms:

• Identify terms with identical variable parts, such as \(-2x^2\) and \(-5x^2\) from the previous expansion.
• Add or subtract coefficients while retaining the common variable and exponent to simplify them into one term.
• For instance, \(-2x^2\) and \(-5x^2\) are combined to \(-7x^2\) due to their shared variable \(x^2\).

After combining all like terms, ensure no further simplifications are possible, resulting in the most streamlined version of the polynomial.