When dealing with algebraic expressions, it is crucial to understand factors and how they work. Factors are numbers or expressions that when multiplied together give a specific result. In this case, we are looking for a binomial that is a factor of the expression 9X2 + 64. Let’s delve into this problem and break it down step by step to find the binomial that is a factor of this expression.
Finding Factors of 9X2 + 64
Before we can determine which binomial is a factor of the expression 9X2 + 64, we need to factor the expression itself. Factoring involves breaking down an expression into its components or simpler forms. In this case, we are looking for factors that will multiply together to give us 9X2 + 64.
Step 1: Identify Perfect Squares
Since 9X2 and 64 are perfect squares, we can apply the formula for factoring the sum of two perfect squares, which is a2 + b2 = (a + b)(a – b). In this formula, a and b represent the perfect squares in the expression.
- 9X2 is a perfect square, which can be factored as (3X)2.
- 64 is a perfect square, which can be factored as 82.
Step 2: Apply the Formula
Now that we have identified the perfect squares in the expression, we can apply the formula for factoring the sum of two perfect squares. The formula states that a2 + b2 = (a + b)(a – b).
Substitute the perfect squares into the formula:
9X2 + 64 = (3X)2 + 82
Using the formula:
(3X + 8)(3X – 8)
Which Binomial Is A Factor of 9X2 + 64
Now that we have factored the expression 9X2 + 64 as (3X + 8)(3X – 8), we can determine which binomial is a factor of this expression. In this case, the binomial that is a factor of 9X2 + 64 is either (3X + 8) or (3X – 8).
Determining the Factors
To find out which binomial is a factor of 9X2 + 64, we can use the division method. We will divide the expression by each binomial and check if the division results in a remainder of zero.
Factor 1: (3X + 8)
Dividing 9X2 + 64 by (3X + 8) gives us:
- Dividend: 9X2 + 64
- Divisor: (3X + 8)
- Quotient: 3X – 8
- Remainder: 0
Since the division of 9X2 + 64 by (3X + 8) results in a zero remainder, we can conclude that (3X + 8) is a factor of the expression.
Factor 2: (3X – 8)
Now, let’s divide 9X2 + 64 by (3X – 8) to see if it is also a factor:
- Dividend: 9X2 + 64
- Divisor: (3X – 8)
- Quotient: 3X + 8
- Remainder: 0
Just like with (3X + 8), the division of 9X2 + 64 by (3X – 8) results in a zero remainder, indicating that (3X – 8) is also a factor of the expression.
Conclusion
In conclusion, the binomials that are factors of the expression 9X2 + 64 are (3X + 8) and (3X – 8). By applying the formula for factoring the sum of two perfect squares, we were able to factor the expression and identify the binomials that divide evenly into it. This understanding of factors and binomials is essential in algebra and lays the foundation for solving various mathematical problems.
