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Magazine Chapter 5: Problem 44
Multiply. See Example 7. $$ (x-2)\left(x^{2}-4 x+2\right) $$
Short Answer
Expert verified
The product is \(x^3 - 6x^2 + 10x - 4\).
Step by step solution
01
Distribute the first term
Start by distributing the first term of the binomial \(x - 2\) to each term in the trinomial. Multiply \(x\) by \(x^2\), \(-4x\), and \(2\). This gives you: \[x \cdot x^2 = x^3\] \[x \cdot (-4x) = -4x^2\] \[x \cdot 2 = 2x\]
02
Distribute the second term
Next, distribute the second term of the binomial, which is \(-2\), to each term in the trinomial. Multiply \(-2\) by \(x^2\), \(-4x\), and \(2\). This gives you: \[-2 \cdot x^2 = -2x^2\] \[-2 \cdot (-4x) = 8x\] \[-2 \cdot 2 = -4\]
03
Combine like terms
Now, write down all the terms from Step 1 and Step 2, and combine like terms: \[x^3 - 4x^2 + 2x - 2x^2 + 8x - 4\] Combine the \(x^2\) terms and the \(x\) terms: \[x^3 + (-4x^2 - 2x^2) + (2x + 8x) - 4\] Simplify: \[x^3 - 6x^2 + 10x - 4\]
04
Final answer
After combining like terms, the simplified expression is: \[x^3 - 6x^2 + 10x - 4\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Distributive Property
The distributive property is a crucial concept in algebra that allows you to multiply a single term across terms inside parentheses. It helps in breaking down expressions to make them more manageable. In the context of our problem, you apply the distributive property to multiply a binomial \(x - 2\) with each term of a trinomial \(x^{2} - 4x + 2\).
- The first task is to distribute the \(x\) from the binomial to each term in the trinomial: \(x \cdot x^2\), \(x \cdot (-4x)\), and \(x \cdot 2\).
- Next, you distribute the \(-2\) to every term in the trinomial: \(-2 \cdot x^2\), \(-2 \cdot (-4x)\), and \(-2 \cdot 2\).
Combining Like Terms
Combining like terms is a step that comes after distributing, and is essential for simplifying expressions. "Like terms" means terms that have the same variable raised to the same power.
- In our example, after distributing, you have terms like \(x^3\), \(-4x^2\), \(-2x^2\), \(2x\), \(8x\), and \(-4\).
- The \(x^2\) terms are similar, so you add them together: \(-4x^2 - 2x^2 = -6x^2\).
- You also combine the \(x\) terms: \(2x + 8x = 10x\).
Binomial
A binomial is a polynomial with exactly two terms. It can appear in various algebraic problems or equations. For our exercise, the binomial is \(x - 2\). Understanding binomials is foundational because they are often multiplied by other polynomials.
- Binomials are straightforward and only involve two parts, but when multiplied with other polynomials, they can lead to complex expressions.
- The basic operation involving binomials is important in higher-level mathematics, as they appear frequently in factorization and equation solving.
Trinomial
A trinomial is a type of polynomial consisting of three terms. In the given exercise, the trinomial is \(x^{2} - 4x + 2\).
- Trinomials typically have a structure like \(ax^2 + bx + c\), and are common in algebraic equations, particularly quadratic equations.
- When multiplying a trinomial by another polynomial like a binomial, it tests one鈥檚 skill in using the distributive property and combining like terms efficiently.
- Trinomials can describe a range of algebraic behaviors depending on the coefficients and signs of their terms.
