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Questions and Answers
What is the primary purpose of factoring polynomials?
What is the primary purpose of factoring polynomials?
- To find the roots of linear equations
- To add complexity to expressions
- To simplify expressions and solve equations (correct)
- To graph functions only
What is the general form of a quadratic expression that can be factored into (x + r)(x + s)?
What is the general form of a quadratic expression that can be factored into (x + r)(x + s)?
- ax^2 - bx + c
- ax^2 + bx - c
- ax^2 + bx + c (correct)
- ax^2 - bx - c
What is the formula for factoring the difference of squares?
What is the formula for factoring the difference of squares?
- a^2 - 2ab + b^2 = (a - b)^2
- a^2 - b^2 = (a + b)(a - b) (correct)
- a^2 + b^2 = (a + b)(a - b)
- a^2 + 2ab + b^2 = (a + b)^2
What is the method of factoring that involves dividing the polynomial into groups of two terms each?
What is the method of factoring that involves dividing the polynomial into groups of two terms each?
What is the purpose of using synthetic division in factoring?
What is the purpose of using synthetic division in factoring?
What is a common mistake to avoid when factoring polynomials?
What is a common mistake to avoid when factoring polynomials?
Flashcards
Factoring polynomials
Factoring polynomials
The process of rewriting a polynomial as a product of simpler polynomials.
Factoring by GCF
Factoring by GCF
Finding the greatest common factor shared by all terms in a polynomial, and then using it to rewrite the polynomial as a product.
Factoring by Grouping
Factoring by Grouping
Dividing a polynomial into pairs of terms, factoring out common binomials from each pair, and then combining the results.
Factoring quadratic expressions
Factoring quadratic expressions
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Factoring by Decomposition
Factoring by Decomposition
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Factoring by Synthetic Division
Factoring by Synthetic Division
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Study Notes
Factoring Polynomials
Definition Factoring polynomials is the process of expressing a polynomial as a product of simpler polynomials.
Why Factor? Factoring is useful for:
- Simplifying expressions
- Solving equations
- Finding roots
- Graphing functions
Methods of Factoring
1. Greatest Common Factor (GCF)
- Find the GCF of all terms in the polynomial
- Divide each term by the GCF
- Write the polynomial as the product of the GCF and the resulting factors
2. Factoring by Grouping
- Divide the polynomial into groups of two terms each
- Factor out the common binomial from each group
- Combine the groups
3. Factoring Quadratic Expressions
- General form:
ax^2 + bx + c - Factoring:
(x + r)(x + s), whererandsare constants randscan be found by solving the equationax^2 + bx + c = 0
4. Factoring by Decomposition
- Express the polynomial as the sum of squares or differences of squares
- Factor using the formulas:
a^2 + 2ab + b^2 = (a + b)^2a^2 - 2ab + b^2 = (a - b)^2
5. Factoring by Synthetic Division
- Divide the polynomial by a linear factor
(x - r) - If the remainder is zero, then
(x - r)is a factor
Common Factoring Mistakes
- Forgetting to factor out the GCF
- Overlooking common binomials
- Not checking for the existence of other factors
Tips and Tricks
- Look for common factors, especially the GCF
- Use the distributive property to expand and simplify
- Check your work by multiplying the factors back together
Factoring Polynomials
Definition and Purpose
- Factoring polynomials is the process of expressing a polynomial as a product of simpler polynomials.
- Factoring is useful for simplifying expressions, solving equations, finding roots, and graphing functions.
Methods of Factoring
Greatest Common Factor (GCF)
- Find the GCF of all terms in the polynomial.
- Divide each term by the GCF.
- Express the polynomial as the product of the GCF and the resulting factors.
Factoring by Grouping
- Divide the polynomial into groups of two terms each.
- Factor out the common binomial from each group.
- Combine the groups to obtain the factored form.
Factoring Quadratic Expressions
- The general form of a quadratic expression is
ax^2 + bx + c. - The factored form is
(x + r)(x + s), whererandsare constants. randscan be found by solving the equationax^2 + bx + c = 0.
Factoring by Decomposition
- Express the polynomial as the sum of squares or differences of squares.
- Use the formulas:
a^2 + 2ab + b^2 = (a + b)^2a^2 - 2ab + b^2 = (a - b)^2to factor.
Factoring by Synthetic Division
- Divide the polynomial by a linear factor
(x - r). - If the remainder is zero, then
(x - r)is a factor.
Common Factoring Mistakes
- Forgetting to factor out the GCF.
- Overlooking common binomials.
- Not checking for the existence of other factors.
Tips and Tricks
- Look for common factors, especially the GCF.
- Use the distributive property to expand and simplify.
- Check your work by multiplying the factors back together.
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