Factor The Following Expression Completely:${ Y(y-3) - 6(y-3) = }$
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Understanding the Problem
In this problem, we are given an algebraic expression in the form of , and we are asked to factor it completely. Factoring an algebraic expression involves expressing it as a product of simpler expressions, called factors. This can help us simplify the expression, make it easier to work with, and even solve equations involving the expression.
Step 1: Identify the Common Factor
The first step in factoring the given expression is to identify any common factors. In this case, we can see that both terms, and , have a common factor of . This means that we can factor out from both terms.
Step 2: Factor Out the Common Factor
Now that we have identified the common factor, we can factor it out from both terms. To do this, we multiply each term by the reciprocal of the common factor, which is . This gives us:
Step 3: Simplify the Expression
Next, we can simplify the expression by combining like terms. In this case, we can combine the two terms with the common factor :
Step 4: Write the Final Factored Form
The final step is to write the expression in its factored form. In this case, we have factored the expression completely, and the final factored form is:
Conclusion
In this problem, we have factored the given algebraic expression completely. We identified the common factor and factored it out from both terms. We then simplified the expression by combining like terms and wrote the final factored form. This process can be applied to any algebraic expression to simplify it and make it easier to work with.
Example Use Case
Factoring an algebraic expression can be useful in a variety of situations. For example, suppose we have an equation involving the expression . By factoring the expression completely, we can simplify the equation and make it easier to solve. This can be particularly useful in solving quadratic equations or other types of equations that involve algebraic expressions.
Tips and Tricks
When factoring an algebraic expression, it's often helpful to look for common factors. This can involve identifying any terms that have a common factor, such as a variable or a constant. By factoring out the common factor, we can simplify the expression and make it easier to work with. Additionally, it's often helpful to use the distributive property to expand the expression and identify any common factors.
Common Mistakes to Avoid
When factoring an algebraic expression, there are several common mistakes to avoid. One common mistake is to forget to factor out the common factor. This can result in an expression that is not simplified and is difficult to work with. Another common mistake is to factor out the wrong factor. This can result in an expression that is not simplified and is difficult to work with. By being careful and taking the time to factor the expression completely, we can avoid these mistakes and simplify the expression effectively.
Real-World Applications
Factoring an algebraic expression has a number of real-world applications. For example, in physics, factoring an expression can be used to simplify equations involving motion or energy. In engineering, factoring an expression can be used to simplify equations involving stress or strain. In economics, factoring an expression can be used to simplify equations involving supply and demand. By being able to factor an algebraic expression, we can simplify complex equations and make them easier to work with.
Conclusion
In conclusion, factoring an algebraic expression is an important skill that can be used in a variety of situations. By identifying common factors and factoring them out, we can simplify the expression and make it easier to work with. This can be particularly useful in solving quadratic equations or other types of equations that involve algebraic expressions. By being careful and taking the time to factor the expression completely, we can avoid common mistakes and simplify the expression effectively.
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Q: What is factoring in algebra?
A: Factoring in algebra involves expressing an algebraic expression as a product of simpler expressions, called factors. This can help us simplify the expression, make it easier to work with, and even solve equations involving the expression.
Q: Why is factoring important in algebra?
A: Factoring is important in algebra because it allows us to simplify complex expressions and make them easier to work with. This can be particularly useful in solving quadratic equations or other types of equations that involve algebraic expressions.
Q: How do I identify common factors in an algebraic expression?
A: To identify common factors in an algebraic expression, look for terms that have a common factor, such as a variable or a constant. You can also use the distributive property to expand the expression and identify any common factors.
Q: What is the distributive property in algebra?
A: The distributive property in algebra is a rule that states that a single term can be distributed to multiple terms inside parentheses. This can be written as:
a(b + c) = ab + ac
Q: How do I factor out a common factor from an algebraic expression?
A: To factor out a common factor from an algebraic expression, multiply each term by the reciprocal of the common factor. This will allow you to factor out the common factor and simplify the expression.
Q: What is the difference between factoring and simplifying an algebraic expression?
A: Factoring an algebraic expression involves expressing it as a product of simpler expressions, called factors. Simplifying an algebraic expression involves combining like terms and eliminating any unnecessary terms.
Q: Can I factor an algebraic expression that has no common factors?
A: Yes, you can factor an algebraic expression that has no common factors. In this case, you can use other factoring techniques, such as factoring by grouping or factoring using the quadratic formula.
Q: How do I factor an algebraic expression that has a quadratic term?
A: To factor an algebraic expression that has a quadratic term, you can use the quadratic formula or factoring by grouping. The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / 2a
Q: What is the quadratic formula?
A: The quadratic formula is a formula that allows you to solve quadratic equations of the form ax^2 + bx + c = 0. It is written as:
x = (-b ± √(b^2 - 4ac)) / 2a
Q: Can I factor an algebraic expression that has a rational term?
A: Yes, you can factor an algebraic expression that has a rational term. In this case, you can use the factoring technique of factoring out the greatest common factor (GCF) of the numerator and denominator.
Q: What is the greatest common factor (GCF) of two numbers?
A: The greatest common factor (GCF) of two numbers is the largest number that divides both numbers evenly.
Q: How do I factor an algebraic expression that has a complex term?
A: To factor an algebraic expression that has a complex term, you can use the factoring technique of factoring out the greatest common factor (GCF) of the real and imaginary parts.
Q: What is a complex term in algebra?
A: A complex term in algebra is a term that has both real and imaginary parts. It is written in the form a + bi, where a and b are real numbers and i is the imaginary unit.
Q: Can I factor an algebraic expression that has a trigonometric term?
A: Yes, you can factor an algebraic expression that has a trigonometric term. In this case, you can use the factoring technique of factoring out the greatest common factor (GCF) of the trigonometric functions.
Q: What is a trigonometric term in algebra?
A: A trigonometric term in algebra is a term that involves trigonometric functions, such as sine, cosine, or tangent.
Q: How do I factor an algebraic expression that has a logarithmic term?
A: To factor an algebraic expression that has a logarithmic term, you can use the factoring technique of factoring out the greatest common factor (GCF) of the logarithmic functions.
Q: What is a logarithmic term in algebra?
A: A logarithmic term in algebra is a term that involves logarithmic functions, such as log or ln.
Q: Can I factor an algebraic expression that has a polynomial term?
A: Yes, you can factor an algebraic expression that has a polynomial term. In this case, you can use the factoring technique of factoring out the greatest common factor (GCF) of the polynomial terms.
Q: What is a polynomial term in algebra?
A: A polynomial term in algebra is a term that involves variables raised to non-negative integer powers, such as x^2 or 3x^4.
Q: How do I factor an algebraic expression that has a rational expression?
A: To factor an algebraic expression that has a rational expression, you can use the factoring technique of factoring out the greatest common factor (GCF) of the numerator and denominator.
Q: What is a rational expression in algebra?
A: A rational expression in algebra is a fraction that involves variables and constants, such as x/y or 3x/2y.
Q: Can I factor an algebraic expression that has a radical term?
A: Yes, you can factor an algebraic expression that has a radical term. In this case, you can use the factoring technique of factoring out the greatest common factor (GCF) of the radical terms.
Q: What is a radical term in algebra?
A: A radical term in algebra is a term that involves a square root or other radical, such as √x or 3√y.
Q: How do I factor an algebraic expression that has a mixed term?
A: To factor an algebraic expression that has a mixed term, you can use the factoring technique of factoring out the greatest common factor (GCF) of the mixed terms.
Q: What is a mixed term in algebra?
A: A mixed term in algebra is a term that involves a combination of variables and constants, such as 2x + 3y or x^2 + 2y.
Q: Can I factor an algebraic expression that has a complex fraction?
A: Yes, you can factor an algebraic expression that has a complex fraction. In this case, you can use the factoring technique of factoring out the greatest common factor (GCF) of the numerator and denominator.
Q: What is a complex fraction in algebra?
A: A complex fraction in algebra is a fraction that involves variables and constants, such as (x + 2)/(x - 3) or (2x + 1)/(x - 2).
Q: How do I factor an algebraic expression that has a nested fraction?
A: To factor an algebraic expression that has a nested fraction, you can use the factoring technique of factoring out the greatest common factor (GCF) of the numerator and denominator.
Q: What is a nested fraction in algebra?
A: A nested fraction in algebra is a fraction that involves a fraction within another fraction, such as (x + 2)/(x - 3) or (2x + 1)/(x - 2).
Q: Can I factor an algebraic expression that has a variable in the denominator?
A: Yes, you can factor an algebraic expression that has a variable in the denominator. In this case, you can use the factoring technique of factoring out the greatest common factor (GCF) of the numerator and denominator.
Q: What is a variable in the denominator in algebra?
A: A variable in the denominator in algebra is a variable that appears in the denominator of a fraction, such as x/y or 3x/2y.
Q: How do I factor an algebraic expression that has a negative exponent?
A: To factor an algebraic expression that has a negative exponent, you can use the factoring technique of factoring out the greatest common factor (GCF) of the numerator and denominator.
Q: What is a negative exponent in algebra?
A: A negative exponent in algebra is an exponent that is less than zero, such as x^(-2) or 3x^(-4).
Q: Can I factor an algebraic expression that has a fractional exponent?
A: Yes, you can factor an algebraic expression that has a fractional exponent. In this case, you can use the factoring technique of factoring out the greatest common factor (GCF) of the numerator and denominator.
Q: What is a fractional exponent in algebra?
A: A fractional exponent in algebra is an exponent that is a fraction, such as x^(1/2) or 3x^(