Problem 71
Question
Factor. If an expression is prime, so indicate. $$ -y^{3}-13 y^{2}-12 y $$
Step-by-Step Solution
Verified Answer
The expression can be factored as
\(-y(y + 1)(y + 12)\).
1Step 1: Identify Common Factors
First, we check if there is a common factor that can be factored out of all the terms. The expression is \(-y^{3} - 13y^{2} - 12y\). Each term has a factor of \(y\). So, factor out \(y\): \[-y(y^2 + 13y + 12)\].
2Step 2: Factor the Quadratic
Now, focus on the quadratic expression inside the parentheses: \(y^2 + 13y + 12\).We need two numbers that multiply to 12 (the constant term) and add up to 13 (the coefficient of the linear term). These numbers are 1 and 12, as:1 \(\times\) 12 = 121 + 12 = 13.Therefore, the quadratic can be factored as:\((y + 1)(y + 12)\).
3Step 3: Write the Complete Factored Expression
Substitute the factored quadratic back into the expression:\[-y(y + 1)(y + 12)\].
4Step 4: Simplify Signs if Necessary
Observe the negative sign with the \(y\) factor that was previously factored out. Simplify the expression to ensure consistent sign convention:\[(-y)(y + 1)(y + 12)\].Ensure that all terms are clearly delineated with the extracted negative sign.
Key Concepts
Quadratic ExpressionGreatest Common FactorFactoring Techniques
Quadratic Expression
A quadratic expression is a type of polynomial expression where the highest degree of the variable is 2. It usually takes the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. Quadratic expressions are commonly encountered in algebra and are central to solving various practical problems.
Among the many interesting properties of quadratic expressions is their graph's shape. Called a parabola, it opens upwards if \(a > 0\) or downwards if \(a < 0\). This makes quadratic expressions important not only in algebra but also in fields like physics and engineering, where relationships between variables often have quadratic forms.
When looking at a quadratic expression, one of the first things to identify is whether it can be factored. Factoring quadratic expressions is a technique that transforms them into products of simpler expressions, which can simplify solving equations. In our exercise, we started with a cubic expression but concentrated on the quadratic portion, \(y^2 + 13y + 12\), to make it more manageable.
To successfully factor a quadratic, we needed numbers that multiply to give the constant term (12 in this case) and add up to the linear coefficient (13 here). This helps in breaking the expression into a product of two simpler expressions.
Among the many interesting properties of quadratic expressions is their graph's shape. Called a parabola, it opens upwards if \(a > 0\) or downwards if \(a < 0\). This makes quadratic expressions important not only in algebra but also in fields like physics and engineering, where relationships between variables often have quadratic forms.
When looking at a quadratic expression, one of the first things to identify is whether it can be factored. Factoring quadratic expressions is a technique that transforms them into products of simpler expressions, which can simplify solving equations. In our exercise, we started with a cubic expression but concentrated on the quadratic portion, \(y^2 + 13y + 12\), to make it more manageable.
To successfully factor a quadratic, we needed numbers that multiply to give the constant term (12 in this case) and add up to the linear coefficient (13 here). This helps in breaking the expression into a product of two simpler expressions.
Greatest Common Factor
The Greatest Common Factor (GCF), also known as the greatest common divisor, is the largest factor that divides two or more numbers. When factoring polynomial expressions, identifying the GCF is crucial as it simplifies the process.
In our original exercise, before focusing on factoring the quadratic expression, we first extracted the GCF of the entire polynomial, \(-y^3 - 13y^2 - 12y\). Since all terms shared a common variable, \(y\), we factored it out initially.
Finding the GCF is a helpful first step because it reduces the given polynomial into a simpler form, making further steps like factoring trinomials easier. By pulling out the common factor, you reduce the work needed when dealing with more complex polynomial expressions. Once the GCF is factored out, only the remaining expression within parentheses is concentrated on for further factoring.
This approach also ensures that the polynomial is simplified as much as possible, presenting the cleanest possible form of the expression.
In our original exercise, before focusing on factoring the quadratic expression, we first extracted the GCF of the entire polynomial, \(-y^3 - 13y^2 - 12y\). Since all terms shared a common variable, \(y\), we factored it out initially.
Finding the GCF is a helpful first step because it reduces the given polynomial into a simpler form, making further steps like factoring trinomials easier. By pulling out the common factor, you reduce the work needed when dealing with more complex polynomial expressions. Once the GCF is factored out, only the remaining expression within parentheses is concentrated on for further factoring.
This approach also ensures that the polynomial is simplified as much as possible, presenting the cleanest possible form of the expression.
Factoring Techniques
Factoring techniques are methods used to break down expressions into simpler multiplicative components. With polynomials, factoring is often key to solving equations, simplifying expressions, and understanding mathematical behavior.
For the quadratic expression \(y^2 + 13y + 12\) in our exercise, we used a specific technique known as "factoring by grouping". This is used when a quadratic is presented in its standard form, and involves finding two numbers that multiply to the constant term (in this case, 12) and add to the coefficient of the linear term (13).
The chosen numbers were 1 and 12. These numbers helped us write the quadratic as \((y + 1)(y + 12)\), turning it into a factorable form. This is a straightforward, effective technique for simple quadratics, though more complex polynomials may require other strategies like completing the square or using the quadratic formula.
Another technique that can be applied is recognizing special patterns, such as the difference of squares, perfect square trinomials, or factoring by substitution. Mastering these techniques is essential as it provides powerful ways to approach and solve a wide variety of algebraic problems, turning complex expressions into more manageable pieces.
For the quadratic expression \(y^2 + 13y + 12\) in our exercise, we used a specific technique known as "factoring by grouping". This is used when a quadratic is presented in its standard form, and involves finding two numbers that multiply to the constant term (in this case, 12) and add to the coefficient of the linear term (13).
The chosen numbers were 1 and 12. These numbers helped us write the quadratic as \((y + 1)(y + 12)\), turning it into a factorable form. This is a straightforward, effective technique for simple quadratics, though more complex polynomials may require other strategies like completing the square or using the quadratic formula.
Another technique that can be applied is recognizing special patterns, such as the difference of squares, perfect square trinomials, or factoring by substitution. Mastering these techniques is essential as it provides powerful ways to approach and solve a wide variety of algebraic problems, turning complex expressions into more manageable pieces.
Other exercises in this chapter
Problem 71
Solve each equation. $$ (2 s-5)(s+6)=0 $$
View solution Problem 71
Explain why \(x^{3}-25\) is not a difference of two cubes.
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Factor each polynomial by factoring out the opposite of the GCF. $$ -4 a^{3} b^{2}+14 a^{2} b^{2}-10 a b^{2} $$
View solution Problem 72
The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 16 d^{2}-5
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