Problem 100 - Elementary Algebra

Question

Factor each trinomial of the form \(x^{2}+b x y+c y^{2}\). \(u^{2}+10 u v+24 v^{2}\)

Step-by-Step Solution

Answer

The factorization of \(u^2 + 10uv + 24v^2\) is \((u + 4v)(u + 6v)\).

Step 1 - Identify the coefficients

Identify the coefficients in the trinomial. Here, the trinomial is in the form of \(u^2 + 10uv + 24v^2\)\, where \(a = 1\), \(b = 10\), and \(c = 24\).

Step 2 - Find factors of the constant term

Find two numbers that multiply to 24 (the constant term) and add up to 10 (the coefficient of the middle term). We need the pairs of factors of 24: \((1, 24)\), \((2, 12)\), \((3, 8)\), \((4, 6)\). Among these, the pair that adds up to 10 is \((4, 6)\).

Step 3 - Write the middle term as a sum

Rewrite the middle term (10uv) using the two numbers found: \(4uv\) and \(6uv\). So the expression becomes \(u^2 + 4uv + 6uv + 24v^2\).

Step 4 - Group and factor

Group the terms into two pairs: \((u^2 + 4uv) + (6uv + 24v^2)\). Factor out the greatest common factor (GCF) from each pair: \(u(u + 4v) + 6v(u + 4v)\).

Step 5 - Factor by grouping

Since \((u + 4v)\) is a common factor, factor it out: \((u + 4v)(u + 6v)\).

Key Concepts

Trinomial Factoring

Trinomial factoring is a crucial concept in algebra that allows us to break down complex expressions into simpler parts. Factoring makes solving equations and simplifying expressions easier.

Let's break down our given problem. We need to factor the trinomial: \(u^2 + 10uv + 24v^2\).

This trinomial fits the form \(ax^2 + bxy + cy^2\) where:

• a is the coefficient of the first term (square term), which here is 1.
• b is the coefficient of the middle term, here it is 10.
• c is the coefficient of the last term (constant term), here 24.

Understanding how to identify these coefficients is the first step towards successfully factoring any trinomial.

Algebraic Expressions

Algebraic expressions consist of variables and constants combined using arithmetic operations.

Consider the expression \(u^2 + 10uv + 24v^2\):

• The variables involved are u and v.
• This expression has three terms.
• Each term can be a product of a constant and variables raised to a power.

Let's rewrite the middle term 10uv into two terms whose product is equal to 24uv (constant term \times\ coefficient of the squared term). We find that 4uv and 6uv work, as:
\[ 4uv \times 6uv = 24uv^2 \text{ and } 4uv + 6uv = 10uv \text{ (the middle term).} \]

So, the expression transforms into: \[ u^2 + 4uv + 6uv + 24v^2. \]

Polynomial Factorization

Polynomial factorization breaks down a polynomial into products of simpler polynomials. Here, we aim to factor \ u^2 + 10uv + 24v^2 \ into its simplest form.

First, group the terms:

• Group 1: \(u^2 + 4uv\)
• Group 2: \(6uv + 24v^2\)

Factor out the greatest common factor (GCF) from each group:

• GCF of \(u^2 + 4uv\) is u: \[ u(u + 4v). \]
• GCF of \(6uv + 24v^2\) is 6v: \[ 6v(u + 4v). \]

Since both groups contain a common binomial factor (u + 4v), factor it out: \[ (u + 4v)(u + 6v). \]

Thus, the trinomial \ u^2 + 10uv + 24v^2 \ is factored into \ (u + 4v)(u + 6v).