Multiply using the Product of Conjugates Pattern.
$(3 x+1)(3 x-1)=$ $\square$
Answer (1)
Identify a and b : In the expression ( 3 x + 1 ) ( 3 x − 1 ) , a = 3 x and b = 1 .
Apply the product of conjugates pattern: ( a + b ) ( a − b ) = a 2 − b 2 .
Substitute a and b : ( 3 x ) 2 − ( 1 ) 2 .
Simplify: ( 3 x ) 2 − ( 1 ) 2 = 9 x 2 − 1 . The final answer is 9 x 2 − 1 .
Explanation
Understanding the Problem We are asked to multiply two binomials using the product of conjugates pattern. The two binomials are ( 3 x + 1 ) and ( 3 x − 1 ) . The product of conjugates pattern is ( a + b ) ( a − b ) = a 2 − b 2 .
Objective We need to multiply ( 3 x + 1 ) ( 3 x − 1 ) using the product of conjugates pattern.
Solution Plan
Identify a and b in the expression ( 3 x + 1 ) ( 3 x − 1 ) . Here, a = 3 x and b = 1 .
Apply the product of conjugates pattern: ( a + b ) ( a − b ) = a 2 − b 2 .
Substitute a = 3 x and b = 1 into the formula: ( 3 x ) 2 − ( 1 ) 2 .
Simplify the expression: ( 3 x ) 2 − ( 1 ) 2 = 9 x 2 − 1 .
The final answer is 9 x 2 − 1 .
Applying the Pattern Let's identify a and b in the expression ( 3 x + 1 ) ( 3 x − 1 ) . In this case, a = 3 x and b = 1 . Now, we apply the product of conjugates pattern, which states that ( a + b ) ( a − b ) = a 2 − b 2 . Substituting a = 3 x and b = 1 into the formula, we get ( 3 x ) 2 − ( 1 ) 2 .
Simplifying the Expression Now, let's simplify the expression. We have ( 3 x ) 2 − ( 1 ) 2 . Squaring 3 x gives us 9 x 2 , and squaring 1 gives us 1 . Therefore, the expression simplifies to 9 x 2 − 1 .
Final Answer The product of ( 3 x + 1 ) ( 3 x − 1 ) using the product of conjugates pattern is 9 x 2 − 1 .
Examples
The product of conjugates pattern is useful in many areas of mathematics, such as simplifying algebraic expressions, rationalizing denominators, and solving equations. For example, consider the expression 2 + 1 1 . To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator, which is 2 − 1 . This gives us 2 + 1 1 × 2 − 1 2 − 1 = 2 − 1 2 − 1 = 2 − 1 . This technique is widely used in calculus and other advanced math courses.
