Problem 155
Question
Factor. \(4 w^{2}-5 w+1\)
Step-by-Step Solution
Verified Answer
\(4w^{2}-5w+1 = (w-1)(4w-1)\)
1Step 1: Identify Terms in the Quadratic
Consider the quadratic expression: \(4w^{2} - 5w + 1\). Identify the coefficients: \(a = 4\), \(b = -5\), and \(c = 1\).
2Step 2: Multiply Coefficient of Quadratic Term and Constant
Multiply the coefficient of the quadratic term \(a\) and the constant term \(c\): \(4 \times 1 = 4\).
3Step 3: Find Factors That Sum to the Middle Term Coefficient
Find the pair of factors of \(4\) that add up to \(-5\). These factors are \(-4\) and \(-1\), since \(-4 + (-1) = -5\).
4Step 4: Rewrite Middle Term Using Found Factors
Rewrite the middle term \( -5w \) as \( -4w - w\) so the expression becomes \( 4w^{2} - 4w - w + 1 \).
5Step 5: Factor by Grouping
Group the terms: \((4w^{2} - 4w) + (-w + 1)\). Factor out the common factors from each group: \(4w(w - 1) - 1(w - 1)\).
6Step 6: Extract Common Binomial Factor
Factor out the common binomial factor \((w - 1)\): \((w - 1)(4w - 1)\).
Key Concepts
quadratic expressionsfactoring by groupingcoefficients
quadratic expressions
Quadratic expressions are an important concept in algebra. They are polynomials that follow the standard form:
ax^2 + bx + c
where `a`, `b`, and `c` are constants, and `a ≠ 0`. In the given exercise, the quadratic expression is: 4w^2 - 5w + 1.
Understanding each part of the quadratic expression can help simplify the factoring process:
- `4w^2` is the quadratic term where the coefficient `a = 4`,
- `-5w` is the linear term with the coefficient `b = -5`, and
- `1` is the constant term `c`.
Recognizing the coefficients makes it easier to apply factoring techniques such as grouping.
ax^2 + bx + c
where `a`, `b`, and `c` are constants, and `a ≠ 0`. In the given exercise, the quadratic expression is: 4w^2 - 5w + 1.
Understanding each part of the quadratic expression can help simplify the factoring process:
- `4w^2` is the quadratic term where the coefficient `a = 4`,
- `-5w` is the linear term with the coefficient `b = -5`, and
- `1` is the constant term `c`.
Recognizing the coefficients makes it easier to apply factoring techniques such as grouping.
factoring by grouping
Factoring by grouping is a strategy used to simplify factoring quadratic expressions. This method involves separating the expression into pairs of terms. Here’s how it works:
Step 1: First, multiply the coefficient `a` of the quadratic term by the constant `c`. In the exercise, this is 4 * 1 = 4.
Step 2: Next, identify two factors of the resulting product that add up to the coefficient `b`. In this case, you need two numbers that multiply to 4 and add up to -5. The factors are -4 and -1.
Step 3: Rewrite the middle term using these factors. Here, -5w becomes -4w - w. This gives you: 4w^2 - 4w - w + 1.
Step 4: Group the expression into two pairs: (4w^2 - 4w) + (-w + 1).
Step 5: Factor out the greatest common factor (GCF) from each group. From the first group, factor out 4w, and from the second group, factor out -1: 4w(w - 1) - 1(w - 1).
Since both groups contain a common binomial factor (w - 1), the expression can be factored as: (w - 1)(4w - 1).
Step 1: First, multiply the coefficient `a` of the quadratic term by the constant `c`. In the exercise, this is 4 * 1 = 4.
Step 2: Next, identify two factors of the resulting product that add up to the coefficient `b`. In this case, you need two numbers that multiply to 4 and add up to -5. The factors are -4 and -1.
Step 3: Rewrite the middle term using these factors. Here, -5w becomes -4w - w. This gives you: 4w^2 - 4w - w + 1.
Step 4: Group the expression into two pairs: (4w^2 - 4w) + (-w + 1).
Step 5: Factor out the greatest common factor (GCF) from each group. From the first group, factor out 4w, and from the second group, factor out -1: 4w(w - 1) - 1(w - 1).
Since both groups contain a common binomial factor (w - 1), the expression can be factored as: (w - 1)(4w - 1).
coefficients
Coefficients are numerical or constant factors of the terms of an algebraic expression. Identifying the coefficients in a quadratic expression is crucial for applying methods like factoring.
Here’s a breakdown of the coefficients in the given exercise:
- **Quadratic Coefficient (`a`)**: This is the number in front of the quadratic term (w^2). In the expression 4w^2, 4 is the coefficient.
- **Linear Coefficient (`b`)**: This is the number in front of the linear term (w). In the exercise, -5w has a coefficient of -5.
- **Constant Term (`c`)**: This is the standalone number without a variable. In the expression, the constant term is 1.
Knowing the coefficients helps to implement factoring methods effectively. For example, in trying to find two numbers that multiply to `ac` and add to `b`, one needs to correctly identify the coefficients to make accurate calculations.
Always start by identifying the coefficients of your expression to use factoring techniques correctly and efficiently.
Here’s a breakdown of the coefficients in the given exercise:
- **Quadratic Coefficient (`a`)**: This is the number in front of the quadratic term (w^2). In the expression 4w^2, 4 is the coefficient.
- **Linear Coefficient (`b`)**: This is the number in front of the linear term (w). In the exercise, -5w has a coefficient of -5.
- **Constant Term (`c`)**: This is the standalone number without a variable. In the expression, the constant term is 1.
Knowing the coefficients helps to implement factoring methods effectively. For example, in trying to find two numbers that multiply to `ac` and add to `b`, one needs to correctly identify the coefficients to make accurate calculations.
Always start by identifying the coefficients of your expression to use factoring techniques correctly and efficiently.