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Chapter 1: Problem 74

Factor the trinomial. $$2(a+b)^{2}+5(a+b)-3$$

Short Answer

Expert verified
The factorization is \((2(a+b) - 1)((a+b) + 3)\).

Step by step solution

01

Identify the Variable Replacement

Notice that the expression contains \((a+b)\) repeated. We can simplify our work by setting \(x = a+b\). This makes our expression \(2x^2 + 5x - 3\).
02

List Potential Factor Pairs

We want to factor \(2x^2 + 5x - 3\) into the form \((mx+n)(px+q)\). The coefficient of the quadratic term is 2 and the constant term is -3. We need factor pairs which when multiplied give us \(2 \times (-3) = -6\). Possible factor pairs of -6 are: (-1, 6), (1, -6), (-2, 3), and (2, -3).
03

Choose the Correct Factor Pair

We need to find a pair that sums up to the middle coefficient, 5. Specifically, solve for \(m \times q + n \times p = 5\): - The correct pair is (-1, 6), leading to the expression: \((2x - 1)(x + 3)\).Verify that this factorization is correct by expanding: \((2x - 1)(x + 3) = 2x^2 + 6x - x - 3 = 2x^2 + 5x - 3\).
04

Substitute Back the Original Variable

We originally set \(x = a + b\). Now that we have factored \(2x^2 + 5x - 3\) into \((2x - 1)(x + 3)\), substitute back \( a + b \). Replace to get \( (2(a+b) - 1)((a+b) + 3) \).

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Key Concepts

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

Variable Substitution
Variable substitution is a helpful technique in simplifying complex algebraic expressions. When presented with an expression that includes repetitive parts, like \(2(a+b)^{2}+5(a+b)-3\), it's beneficial to substitute the repeating element with a single variable.
This step transforms a potentially complicated expression into a simpler form. For example:
  • Identify the repetitive expression, \(a + b\), in this exercise.
  • Substitute with a new variable, say \(x = a + b\).
This changes the equation to \(2x^2 + 5x - 3\), a standard quadratic formula we can easily work with. Simplifying the expression in this way allows us to focus on familiar methods of solving quadratic equations without getting tangled in the complexity of the original variables.
Quadratic Trinomials
Quadratic trinomials are polynomial expressions of the form \(ax^2 + bx + c\). The key to working with these is to factor them, which means expressing them as a product of two binomials. In the exercise, \(2x^2 + 5x - 3\) is the quadratic trinomial.
To factor a quadratic trinomial:
  • Examine the coefficients: here, 2, 5, and -3.
  • Multiply the leading coefficient (2) and the constant (-3) to form -6.
You then find two numbers that multiply to give -6 and add to give the middle coefficient, 5. These factor pairs are crucial in uncovering the relationship between the terms of the trinomial. The correct numbers you select aid in breaking the middle term into parts that can be grouped and factored into binomials.
Factorization Verification
After attempting to factorize a quadratic trinomial, it鈥檚 critical to verify the solution. Verification ensures that the factorization is correct. In the exercise, we deduced the factorization of \(2x^2 + 5x - 3\) as \((2x - 1)(x + 3)\).
To verify:
  • Expand the factorized form to see if it matches the original expression.
  • Use the distributive property: \( (2x - 1)(x + 3) = 2x^2 + 6x - x - 3\).
  • Simplify it to \(2x^2 + 5x - 3\), confirming it equals the initial trinomial.
Verification like this ensures reliability and correctness in algebraic transformations. Once confirmed, substitute back the original variable, \(x = a + b\), to return to the initial form: \((2(a+b) - 1)((a+b) + 3)\). This reinforces the understanding of variable substitution and solving quadratic trinomials.

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