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Questions and Answers
If f(x) = x^3 - 2x^2 - 5x + 6 and f(x) is divided by (x - 1), what is the remainder?
Which of the following is a factorization of x^2 + 5x + 6?
What is the result of (x^2 + 2x - 1) - (x^2 - 3x - 2)?
If (x + 2)(x - 3) = x^2 - x - 6, what is the product of (x + 2) and (x - 3)?
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What is the result of (2x^2 + 3x - 1) + (x^2 - 2x - 3)?
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If f(x) = x^2 + 2x - 3 and f(a) = 5, what is the value of a?
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Study Notes
Polynomials
Addition of Polynomials
- To add two or more polynomials, combine like terms
- Add corresponding coefficients
- Example:
(2x^2 + 3x - 1) + (x^2 - 2x - 3) = 3x^2 + x - 4
Subtraction of Polynomials
- To subtract one polynomial from another, change the sign of the polynomial being subtracted and add
- Example:
(2x^2 + 3x - 1) - (x^2 - 2x - 3) = x^2 + 5x + 2
Multiplication of Polynomials
- To multiply two polynomials, multiply each term of one polynomial by each term of the other
- Combine like terms
- Example:
(2x + 3)(x + 2) = 2x^2 + 7x + 6
Factorization of Polynomials
- Factorization: expressing a polynomial as a product of simpler expressions
- Methods:
- Factoring out greatest common factor (GCF)
- Factoring quadratic expressions (e.g., difference of squares)
- Factoring sum or difference of cubes
- Example:
x^2 + 5x + 6 = (x + 3)(x + 2)
Remainder Theorem
- If a polynomial
f(x)is divided by(x - a), the remainder isf(a) - Example: if
f(x) = x^2 + 2x - 3, thenf(2) = 2^2 + 2(2) - 3 = 5, which is the remainder when dividingf(x)by(x - 2) - The Remainder Theorem is useful for evaluating polynomials and finding roots.
Polynomials
Addition of Polynomials
- Combine like terms to add two or more polynomials
- Add corresponding coefficients to get the resulting polynomial
Subtraction of Polynomials
- Change the sign of the polynomial being subtracted and add to get the resulting polynomial
- Subtracting a polynomial is equivalent to adding its negative
Multiplication of Polynomials
- Multiply each term of one polynomial by each term of the other to get the product
- Combine like terms to simplify the resulting polynomial
Factorization of Polynomials
- Factorization is expressing a polynomial as a product of simpler expressions
- Methods of factorization include:
- Factoring out the greatest common factor (GCF)
- Factoring quadratic expressions (e.g., difference of squares)
- Factoring sum or difference of cubes
- Factorization can be used to simplify polynomials and solve equations
Remainder Theorem
- The remainder of dividing a polynomial
f(x)by(x - a)isf(a) - The Remainder Theorem is useful for evaluating polynomials and finding roots
- The theorem can be used to find the remainder of dividing a polynomial by a linear factor
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Description
This quiz covers the basics of polynomial operations, including addition, subtraction, and multiplication of polynomials.
