2. (1.9) True or False: To factor [tex]$x^2-13 x+30$[/tex], you need to find two integers that multiply to -13 and add to 30. Explain your answer.
3. (1.10)
a. State the difference of squares pattern
b. Use the difference of squares pattern to factor [tex]$x^2-36$[/tex]
4. (1.10)
a. State the two binomial squares patterns.
b. Use the binomial squares pattern to factor [tex]$x^2-12 x+36$[/tex].
5. (1.10) The sum of cubes pattern [tex]$a^3+b^3$[/tex] factors to [tex]$(a+b)(a^2-a b+b^2)$[/tex]. Which of the polynomials below would be a sum of cubes? Factor it.
i. [tex]$x^2+25$[/tex]
ii. [tex]$y^3+125$[/tex]
iii. [tex]$x^3+x^2+8$[/tex]
Answer (2)
The statement about factoring x 2 − 13 x + 30 is false as the two integers must multiply to 30 and add to -13. The differences of squares and two binomial squares patterns are explained and applied to relevant expressions, demonstrating their factorizations. The polynomial y 3 + 125 is identified as a sum of cubes and factored accordingly.
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The statement about factoring x 2 − 13 x + 30 is false because the numbers must multiply to 30 and add to -13.
The difference of squares pattern is a 2 − b 2 = ( a − b ) ( a + b ) , and x 2 − 36 factors to ( x − 6 ) ( x + 6 ) .
The binomial squares patterns are ( a + b ) 2 = a 2 + 2 ab + b 2 and ( a − b ) 2 = a 2 − 2 ab + b 2 , and x 2 − 12 x + 36 factors to ( x − 6 ) 2 .
y 3 + 125 is a sum of cubes and factors to ( y + 5 ) ( y 2 − 5 y + 25 ) .
Explanation
Analyzing the Factoring Statement The statement is False . To factor a quadratic expression in the form x 2 + b x + c , you need to find two numbers that multiply to c (the constant term) and add up to b (the coefficient of the x term). In this case, you need two numbers that multiply to 30 and add to -13.
Stating the Difference of Squares Pattern The difference of squares pattern states that for any two terms a and b , the difference of their squares can be factored as follows: a 2 − b 2 = ( a − b ) ( a + b ) This pattern is very useful for quickly factoring expressions in this form.
Factoring Using Difference of Squares To factor x 2 − 36 using the difference of squares pattern, we recognize that x 2 is the square of x and 36 is the square of 6 (since 6 2 = 36 ). Thus, we can apply the pattern with a = x and b = 6 :
x 2 − 36 = x 2 − 6 2 = ( x − 6 ) ( x + 6 ) So, the factored form of x 2 − 36 is ( x − 6 ) ( x + 6 ) .
Stating Binomial Squares Patterns The two binomial squares patterns are:
The square of a sum: ( a + b ) 2 = a 2 + 2 ab + b 2
The square of a difference: ( a − b ) 2 = a 2 − 2 ab + b 2 These patterns help to quickly expand or factor perfect square trinomials.
Factoring Using Binomial Squares Pattern To factor x 2 − 12 x + 36 using the binomial squares pattern, we look for a pattern of the form a 2 − 2 ab + b 2 . We see that x 2 is the square of x , and 36 is the square of 6 . Also, the middle term is − 12 x , which is − 2 times x times 6 . Thus, we can apply the pattern with a = x and b = 6 :
x 2 − 12 x + 36 = x 2 − 2 ( x ) ( 6 ) + 6 2 = ( x − 6 ) 2 So, the factored form of x 2 − 12 x + 36 is ( x − 6 ) 2 .
Identifying and Factoring Sum of Cubes We are given the sum of cubes pattern: a 3 + b 3 = ( a + b ) ( a 2 − ab + b 2 ) . We need to identify which of the given polynomials is a sum of cubes. i. x 2 + 25 is a sum of squares, not cubes. ii. y 3 + 125 = y 3 + 5 3 is a sum of cubes, since 125 = 5 3 .
iii. x 3 + x 2 + 8 is not a sum of cubes. Now, we factor y 3 + 125 using the sum of cubes pattern with a = y and b = 5 :
y 3 + 125 = y 3 + 5 3 = ( y + 5 ) ( y 2 − 5 y + 25 ) So, the factored form of y 3 + 125 is ( y + 5 ) ( y 2 − 5 y + 25 ) .
Examples
Factoring polynomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to simplify equations when designing structures, ensuring stability and efficiency. Similarly, economists use factoring to analyze market trends and predict future outcomes. Understanding factoring helps in problem-solving across various fields, making complex calculations more manageable and providing valuable insights.
