Problem 22 Multiply and, if possible, simpl... [FREE SOLUTION]

Chapter 7: Problem 22

Multiply and, if possible, simplify. $$(n-3) \cdot \frac{n^{2}+4 n}{n^{2}-5 n+6}$$

Short Answer

Expert verified

\[\frac{n(n + 4)}{n - 2}\]

Step by step solution

Factor the denominator

First, factor the denominator of the fraction: \[n^2 - 5n + 6\]This quadratic expression can be factored into:\[(n-2)(n-3)\]

Rewrite the expression

Rewrite the given expression by replacing the denominator with its factored form:\[(n-3) \times \frac{n^2 + 4n}{(n-2)(n-3)}\]

Simplify the fraction

Since $n-3$ in the numerator and the denominator cancel each other out, simplify the expression to:\[\frac{n^2 + 4n}{n-2}\]

Final answer

The simplified expression is: \[\frac{n(n + 4)}{n - 2}\]Here, $n^2 + 4n$ is rewritten as $n(n+4)$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

factoring quadratics

When dealing with algebraic fractions, one of the key techniques is factoring quadratics. Consider the quadratic expression in the denominator: To factor this, we look for two numbers whose product is 6 and whose sum is -5. These numbers are -2 and -3. Thus, we can rewrite the expression as: Factoring quadratics can often be tricky, so remember: always start by identifying the product and sum necessary to create the middle term.

simplifying expressions

Simplifying expressions in algebra often means factoring, canceling common terms, and reducing fractions. After factoring the quadratic expression in the denominator: Notice that both the numerator and denominator share a common factor Cancel out this common factor to simplify the expression: It's important to always look for common factors in the numerator and denominator when simplifying expressions, as this will often allow you to reduce complex fractions into simpler forms.

polynomial multiplication

Polynomial multiplication involves multiplying each term in one polynomial by each term in another polynomial. In this case, we multiply the polynomial by the factored quadratic expression in the denominator: We then rearrange the expression: It's crucial to understand polynomial multiplication as it forms the foundation for more advanced algebraic operations. Make sure to carefully follow each step and recheck your work, ensuring no term is missed and proper simplifications are made.

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91影视

Multiply and, if possible, simplify. $$(n-3) \cdot \frac{n^{2}+4 n}{n^{2}-5 n+6}$$

Short Answer

Step by step solution

Factor the denominator

Rewrite the expression

Simplify the fraction

Final answer

Key Concepts

factoring quadratics

simplifying expressions

polynomial multiplication

One App. One Place for Learning.

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