Problem 30 - Elementary Algebra

Question

Factor. $$ a 2 b 2+5 a b-50 $$

Step-by-Step Solution

Answer

The expression factors to \((ab + 10)(ab - 5)\).

Step 1: Identify the Expression

The given expression is a quadratic trinomial \(a^2b^2 + 5ab - 50\). We need to factor this expression.

Step 2: Rewrite the Expression with Substitution

To simplify factoring, recognize that the expression can be seen as a quadratic in terms of \(ab\). Let \(x = ab\). Substitute \(x\) into the expression: \(x^2 + 5x - 50\).

Step 3: Factor the Quadratic Expression

Now, factor the quadratic expression \(x^2 + 5x - 50\). We are looking for two numbers that multiply to \(-50\) and add up to \(5\). These numbers are \(10\) and \(-5\) so that \((x+10)(x-5) = x^2 + 5x - 50\).

Step 4: Substitute Back the Original Variable

Now replace \(x\) back with \(ab\). The factored expression becomes \((ab + 10)(ab - 5)\).

Step 5: Verification

Verify the factorization by expanding \((ab + 10)(ab - 5)\) to ensure it equals the original expression. Expand: \((ab + 10)(ab - 5) = a^2b^2 - 5ab + 10ab - 50 = a^2b^2 + 5ab - 50\), which matches the original. The factorization is correct.

Key Concepts

Quadratic Trinomial

A quadratic trinomial is a polynomial composed of three terms where the highest degree of any variable is squared. In our problem, the expression given is \(a^2b^2 + 5ab - 50\). Here, the highest degree is determined by the term \(a^2b^2\), where both variables \(a\) and \(b\) are squared. Quadratic trinomials usually come in the form \(ax^2 + bx + c\). It’s essential to identify it as a quadratic so we can apply specific techniques for simplification and factoring.
Recognizing the expression as a quadratic sets the stage for effective factorization. This understanding paves the way for the next step, which is making the factorization process manageable.

Factorization Process

The factorization process involves breaking down a quadratic trinomial into two simpler binomial expressions. Once we identify something as a quadratic equation, we aim to rewrite it in a factored form of \((mx + n)(px + q)\). For our expression \(x^2 + 5x - 50\), derived through substitution, it becomes crucial to find two numbers whose product equals \(-50\) (the constant term) and sum equals \(5\) (the coefficient of the linear term). In this case, those numbers are \(10\) and \(-5\).
This matches the pattern of successful factorization, where the middle coefficient changes from addition to subtraction when combining terms. The factorization \((x+10)(x-5)\) represents these relationships perfectly.

Variable Substitution

Variable substitution is a technique used to simplify complex expressions involving multiple variables by replacing them with a single variable. In our case, we let \(x = ab\), turning our expression into \(x^2 + 5x - 50\). This substitution reduces the complexity, turning a multi-variable polynomial into a more straightforward quadratic equation.

• Turning the expression into a single variable form makes factorization easier.
• Focuses the complexity on simpler arithmetic and algebraic operations.
• Allows us to apply standard techniques for factoring quadratics.

Once factorized, this makes it effortless to substitute back to the original variables, maintaining the integrity of the initial expression.

Verification of Factorization

Verification of factorization is a critical step in solving polynomial expressions. We must ensure that our factors are correctly derived by expanding the factors back into the original polynomial. For our expression, verifying \((ab + 10)(ab - 5)\) requires expanding by applying the distributive property: \((ab + 10)(ab - 5) = a^2b^2 - 5ab + 10ab - 50 = a^2b^2 + 5ab - 50\).

• This confirms the factorization matches the original trinomial.
• Detects any errors in the factorization process.
• Ensures the solution's correctness and reliability.

Verification not only proves the solution is correct, but it also builds confidence in handling factorization problems.