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Questions and Answers
Which of the following trinomials can be factored into two binomials with a positive constant term and a negative coefficient for the $x$ term?
Which of the following trinomials can be factored into two binomials with a positive constant term and a negative coefficient for the $x$ term?
- $x^2 - 7x - 12$
- $x^2 - 7x + 12$ (correct)
- $x^2 +7x + 12$
- $x^2 + 7x - 12$
If the factors of a trinomial have a product of 15 and a sum of 8, what is the trinomial?
If the factors of a trinomial have a product of 15 and a sum of 8, what is the trinomial?
- $x^2 - 8x + 15$
- $x^2 + 8x + 15$ (correct)
- $x^2 - 8x - 15$
- $x^2 + 8x - 15$
What is the incorrect statement about the factors of the trinomial $x^2 - 5x - 14$?
What is the incorrect statement about the factors of the trinomial $x^2 - 5x - 14$?
- The factors are $(x-7)$ and $(x+2)$.
- The factors are both positive. (correct)
- The factors have a product of -14.
- The factors have a sum of -5.
A student claims that the trinomial $x^2 + 14x + 48$ can be factored into $(x+8)$ and $(x-6)$. Which statement correctly identifies the student's error?
A student claims that the trinomial $x^2 + 14x + 48$ can be factored into $(x+8)$ and $(x-6)$. Which statement correctly identifies the student's error?
Which statement about trinomials with factors that have a negative product is true?
Which statement about trinomials with factors that have a negative product is true?
Flashcards
Factoring Polynomials
Factoring Polynomials
The process of breaking down a polynomial into its factor pairs.
Correct Factor Pair Example
Correct Factor Pair Example
An example where a polynomial matches its factors, e.g., $x^2 + 2x - 63$ has factors $(x+9)$ and $(x-7)$.
Incorrect Statement Identification
Incorrect Statement Identification
Finding incorrect claims about a polynomial's factors, like for $x^2 - 12x + 32$.
Matching Trinomials to Students
Matching Trinomials to Students
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Factors' Signs and Product/Sum
Factors' Signs and Product/Sum
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Study Notes
Factoring Polynomials
- Polynomials can be factored into simpler expressions
- A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
- Factoring involves finding the factors which when multiplied gives the actual expression.
- Different methods exist to factor different types of polynomials, such as Trinomials
Trinomial Factoring
- To factor a trinomial of the form ax² + bx + c, find two numbers that multiply to ac and add to b
- For instance, if we have x² + 5x - 24, we look for two numbers that multiply to -24 and add to 5. These numbers are 8 and -3
- The factored form in this case is (x + 8)(x - 3)
Example Polynomials to Factor
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8x - 65 : This is a binomial, not a trinomial, so this cannot be factored using the method described here.
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23x + 60: This is a binomial, not a trinomial, so this cannot be factored using the method described here.
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x² - 12x + 32: Factoring this trinomial, we look for two numbers that multiply to 32 and add to -12. These two numbers would be (-8 and -4), therefore factorization is (x-8)(x-4).
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x² + 15x + 44: Two numbers that multiply to 44 and add to 15 are 11 and 4. Factoring, we get (x+11)(x+4).
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6x² - 72: Factoring shows it is 6(x-2 * (x+2))
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x² + 5x - 84: Factoring shows it is (x+12)(x-7)
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x² + 6x - 27: Factoring shows it is (x+9)(x-3)
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x² - 14x + 33: Factoring shows it is (x-11)(x-3)
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x² + 5x - 84: Factoring this trinomial, we look for two numbers that multiply to -84 and add to 5. These numbers are 12 and -7, therefore factorization is (x+12)(x-7).
Identifying Incorrect Statements
- Students might make errors in the way they factor
- Pay attention to the sign of the factors when determining the correctness of the factored form
- The product and sum should be correctly calculated
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