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Magazine Chapter 6: Problem 55
Factor each trinomial completely. See Examples I through II and Section 6.2 \(3 a^{2}+10 a b+3 b^{2}\)
Short Answer
Expert verified
The trinomial factors to \((3a + b)(a + 3b)\).
Step by step solution
01
Identify the trinomial structure
The given trinomial is \(3a^2 + 10ab + 3b^2\). It follows the general form \(ax^2 + bx + c\), where \(a = 3\), \(b = 10\), and \(c = 3\). In this case, the variables involve both \(a\) and \(b\). Our goal is to express this trinomial as a product of two binomials.
02
Find two numbers that multiply to ac and add to b
Calculate \(ac\), where \(a = 3\) and \(c = 3\), resulting in \(ac = 9\). We need two numbers that multiply to 9 and add up to 10 (the coefficient of \(ab\)). The numbers 9 and 1 satisfy both conditions: \(9 \times 1 = 9\) and \(9 + 1 = 10\).
03
Rewrite the middle term using the two numbers found
Rewrite the trinomial as four terms by splitting the middle term \(10ab\) into \(9ab\) and \(1ab\). The expression now becomes: \(3a^2 + 9ab + ab + 3b^2\).
04
Factor by grouping
Group the terms into two pairs: \((3a^2 + 9ab)\) and \((ab + 3b^2)\). Then factor each pair individually. The first pair can be factored as \(3a(a + 3b)\) and the second as \(b(a + 3b)\).
05
Factor out the common binomial factor
Both groups contain the common binomial factor \((a + 3b)\). Factor this binomial out to obtain the expression: \((3a + b)(a + 3b)\). Thus, the trinomial \(3a^2 + 10ab + 3b^2\) is factored completely as \((3a + b)(a + 3b)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. These expressions form the building blocks of algebra. For instance, in the exercise you encountered, the expression \(3a^2 + 10ab + 3b^2\) is an algebraic expression consisting of variables and terms.
- **Terms**: In algebra, terms are the individual components of an expression separated by addition (+) or subtraction (-) signs.
- **Variables and Coefficients**: Variables are symbols like \(a\) and \(b\) that represent numbers, while the numbers multiplying them are called coefficients, like 3 in \(3a^2\).
- **Expressions and Equations**: Unlike an equation, an expression does not have an equals sign. It's like an incomplete sentence that needs to be solved or simplified.
Polynomial Factoring
Polynomial factoring involves breaking down a polynomial, like the trinomial in the exercise, into simpler 'factor' polynomials that, when multiplied, give back the original polynomial. It's a key skill in algebra for simplifying expressions and solving equations.
Here's how you can approach polynomial factoring:
Here's how you can approach polynomial factoring:
- **Identify the Polynomial**: Begin by evaluating the polynomial's form. Verify if it's a trinomial, as in the example \(3a^2 + 10ab + 3b^2\), which is of the form \(ax^2 + bx + c\).
- **Find Factor Pairs**: Determine pairs of numbers that multiply to \(ac\) and add to \(b\). This identifies the factors that are used later in factoring by grouping.
- **Factor by Grouping**: This involves rearranging terms and grouping them to extract common factors easily.
Quadratic Trinomials
Quadratic trinomials are polynomials consisting of three terms with a variable raised to the second power. The general form is \(ax^2 + bx + c\), which can be expressed in terms of variables like \(a\) and \(b\) seen in the example.
A focus on quadratic trinomials includes:
A focus on quadratic trinomials includes:
- **Quadratic Term**: This is the term \(ax^2\) where \(x\) is raised to the second power, crucial in determining the standard form of quadratics.
- **Middle and Constant Terms**: The middle term \(bx\) and constant \(c\) need to align perfectly for the trinomial to be factored into binomials effectively.
- **Trinomial Factoring**: Often, quadratics require determining two binomials whose product returns the original trinomial.
