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Magazine Chapter 1: Problem 68
Factor the trinomial. $$x^{2}-6 x+5$$
Short Answer
Expert verified
The trinomial factors as \((x - 5)(x - 1)\).
Step by step solution
01
Identify the Trinomial
First, identify and write down the trinomial: \[x^2 - 6x + 5\]This is a quadratic trinomial where the coefficient of \(x^2\) is 1, the coefficient of \(x\) is -6, and the constant term is 5.
02
Find Two Numbers That Multiply and Add
We need two numbers that multiply to the constant term 5 and add to the coefficient of \(x\), which is -6. - The pair (1, 5) multiplies to 5, but doesn't add to -6.- The pair (-1, -5) multiplies to 5 and adds to -6.
03
Write the Middle Term as a Sum
Rewrite the middle term, -6x, as a sum of two terms using the numbers found:\[x^{2} - 6x + 5 = x^{2} - 1x - 5x + 5\]This step expresses -6x as -1x and -5x.
04
Group Terms and Factor Out Common Factors
Group the terms into two pairs and factor each group:\[x^{2} - 1x - 5x + 5 = (x^{2} - 1x) + (-5x + 5)\]Factor out \(x\) from the first group and \(-5\) from the second group:\[x(x - 1) - 5(x - 1)\]
05
Factor by Grouping
Notice that each group includes the common factor \((x - 1)\):\[x(x - 1) - 5(x - 1) = (x - 5)(x - 1)\]Factor \((x - 1)\) out to get the final factored form.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Equations
A quadratic equation is a type of polynomial equation of the form \(ax^2 + bx + c = 0\). It is characterized by having a degree of 2, which means the highest exponent of the variable \(x\) is 2. Quadratic equations are commonplace in various real-world scenarios, and understanding how to solve them is a fundamental math skill. In our example, the quadratic equation is \(x^2 - 6x + 5 = 0\). This showcases a common type where the leading coefficient, \(a\), is equal to 1.To solve quadratic equations, you can use several methods:
- Factoring: Splitting the equation into two binomial expressions that multiply to give the original trinomial.
- Completing the square: Adjusting the equation to express it as a perfect square trinomial.
- Quadratic formula: A universal formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), useful especially when factoring is challenging.
Polynomial Factoring
Polynomial factoring involves breaking down a polynomial into simpler "factor" polynomials. This makes solving equations easier. In essence, factoring transforms a polynomial into the product of two or more polynomials. For example, the polynomial \(x^2 - 6x + 5\) is rewritten as \((x - 5)(x - 1)\) in its factored form.Here's a step-by-step on how it works:
- Identify and separate: Write down the trinomial clearly.
- Find pairs: Look for two numbers that multiply to the last term (here 5) and add up to the middle coefficient (-6).
- Replace and group: Use these numbers to split the middle term, and group terms.
- Factor out: Identify and factor out the greatest common factor in each group.
- Combine common factors: Simplify the expression by combining the common factors found.
Completing the Square
Completing the square is another method to solve quadratic equations. It's particularly beneficial for converting a quadratic equation into a form that is much easier to solve. The method involves adjusting the equation so that it forms a perfect square trinomial, allowing it to be written as \((x - p)^2 = q\).The steps for completing the square are:
- Move the constant term: Arrange the quadratic equation so the constant term is on the opposite side.
- Divide the middle coefficient: Take half of the coefficient of \(x\), square it, and add it to both sides to maintain the equation's equality.
- Form the perfect square trinomial: The equation should now look like a squared binomial.
- Solve for \(x\): Take the square root of both sides and solve for \(x\).
